Minkowski functional

In mathematics, in the field of functional analysis, a Minkowski functional is a function that recovers a notion of distance on a linear space.

If K subset of a real or complex vector space X, then we define the Minkowski functional or gauge of K to be the function p_K : X → [0, ∞], valued in the extended real numbers, defined by

p_K(x) := inf { r ∈ ℝ : r > 0 and x ∈ rK }

for every x ∈ X, where recall the infimum of the of the empty set is defined to be positive infinity ∞ (which is not a real number so that p_K(x) would then not be real-valued). If the set is not empty then the infimum will necessarily be a non-negative real number. This property of being non-negative stands in contrast to other classes of functions, such as sublinear functions, that do allow negative values.

In functional analysis, K is usually assumed to have properties (e.g. such as being absorbing in X) that will guarantee that for every x ∈ X, this set { r ∈ ℝ : r > 0 and x ∈ rK } is not empty precisely because this results in p_K being real-valued.

Moreover, K is also often assumed to have more properties, such as being an absorbing disk in X, since these properties guarantee that p_K will be a (real-valued) seminorm on X. In fact, every seminorm p on X is equal to the Minkowski functional of any subset K of X satisfying { x ∈ X : p(x) < 1 } ⊆ K ⊆ { x ∈ X : p(x) ≤ 1 } (where all three of these sets are necessarily absorbing in X and the first and last are also disks). Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis. In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of X into certain algebraic properties of a function on X.

Definition

Given a subset K of a real or complex vector space X, one can define the gauge of K or the Minkowski functional associated with or induced by K as being the function p_K : X → [0, ∞], valued in the extended real numbers, defined by

p_K(x) := inf { r > 0 : x ∈ rK },

where recall that the infimum of the empty set is ∞ (i.e. inf ∅ = ∞). Here, { r > 0 : x ∈ rK } is shorthand for { r ∈ ℝ : r > 0 and x ∈ rK }. Observe that for any x ∈ X, p_K(x) ≠ ∞ if and only if { r > 0 : x ∈ rK } is not empty.

Recall that we may partially extend ℝ's arithmetic operations to include ±∞, where r/±∞ := 0 for all non-zero real -∞ < r < ∞. Note that 0 ⋅ ∞ and 0 ⋅ -∞ remain undefined.

Some conditions making a gauge real-valued

In the field of convex analysis, the map p_K taking on the value of ∞ is not necessarily an issue. However, in functional analysis we almost always want p_K to be real-valued (i.e. to never take on the value of ∞), which happens if and only if the set { r > 0 : x ∈ rK } is non-empty for every x ∈ X.

In order for p_K to be real-valued, it suffices for the origin of X to belong to the algebraic interior (or core) of K in X.[1] If K is absorbing in X, where recall that this implies that 0 ∈ K, then the origin belongs to the algebraic interior of K in X and thus p_K is real-valued. Characterizations of when p_K is real-valued are given below.

Properties (minimal requirements)

In this section, we will investigate the most general case of the gauge of any subset K of X. The more common special case where K is assumed to be an absorbing disk in X is discussed after this section. However, all results in this section may be applied to the case where K is an absorbing disk.

Notation

As before, let K be any subset of a real or complex vector space X.

We will make heavy use the following types of sets (which are defined as usual) to characterize or deduce properties of Minkowski functionals.

Notation: If A is any set of scalars then let AK := { ak : a ∈ A, k ∈ K }. Thus for any real R and S, (R, S)K := { tk : R < t < S, k ∈ K }, and [R, S)K := { tk : R ≤ t < S, k ∈ K }, and we define the sets (R, S]K, (R, ∞)K, etc. similarly. To simplify the discussion, we also define (R, ∞]K := (R, ∞)K and [R, ∞]K := [R, ∞)K. Also, if I is any such interval or set of scalars then for every x ∈ X, we will let Ix denote I { x }; for instance (R, S)x := { tx : R < t < S }.

Examples

• Let X = ℝ and x ∈ X. If x > 0 then (0, 1)x is just the interval (0, x); that is, it is the open line segment lying strictly between the origin and x. Similarly, if x > 0 then (1, ∞)x = (x, ∞) is the infinite open ray starting at (but not including) x and heading directly away from the origin. If x = 0 then (0, 1)x = { 0 } = (1, ∞)x = ℝ x. If x < 0 then (0, x) = ∅ and (0, 1)x = (x, 0).
• (0, 1) K = ∪k ∈ K (0, 1) k is the union of all open line segments lying strictly between the origin and some k ∈ K (except if k = 0 ∈ K, in which case (0, 1) k = { 0 }).
• Let x ∈ X, C := [0, 1] x, L := [0, 1) x, and R := (0, 1] x. If I is any of the intervals [0, 1], [0, 1), (0, 1], or (0, 1) then L = I L, C ≠ I L, R ≠ I L, and R ≠ I C. Also, L = [0, 1) C = (0, 1) C so in particular, the last two sets are not equal to C; moreover, if x ≠ 0 then also C ≠ (0, 1) C. However, we do have C = (0, 1] C = [0, 1] C and R = (0, 1] R = [0, 1] R.

Properties

If A is a set of scalars, -∞ ≤ R ≤ S ≤ ∞, and x ∈ X then we will use the following basic properties without comment:

• Inclusion in a set: x ∈ (R, S)K if and only if tx ∈ (tR, tS)K for all real t > 0. Moreover, x ∈ AK if and only if sx ∈ AsK for all scalars s.
• Exclusion from a set: x ∉ (R, S)K if and only if tx ∉ (tR, tS)K for all real t > 0; if neither R nor S is infinite then we may replace "t > 0" with "t ≥ 0" in the last condition. Moreover, x ∉ AK if and only if sx ∉ AsK for all non-zero scalars s.
  • Note that in contrast to inclusion, sx ∉ AsK for all scalars s if and only if AK = ∅.

Seminorms and Minkowski functionals

The following corollary mainly summarizes and combines the results that are established in the basic properties section below.

Corollary and Summary — Suppose that K is a subset of a real or complex vector space X.

1. Subadditive/Triangle inequality: p_K is subadditive if and only if (0, 1) K is convex. If K is convex then so are (0, 1) K and (0, 1] K and p_K is subadditive.
2. Real-valued: p_K is real-valued if and only if (0, ∞)K = X, in which case 0 ∈ K.
3. Absorbing: If K is convex (or balanced) and if (0, ∞)K = X then K is absorbing in X.
   • Note that if a set A is absorbing in X and A ⊆ K then K is absorbing in X.
   • If K is convex and 0 ∈ K then [0, 1] K = K, so in particular (0, 1) K ⊆ K.
4. Strict positive homogeneity: p_K(rx) = r p_K(x) for all x ∈ X and all positive real r > 0.
5. Positive/Nonnegative homogeneity: p_K is nonnegative homogeneous if and only if p_K is real-valued.
6. Absolute homogeneity: p_K(u x) = p_K(x) for all x ∈ X and all unit length scalars u (i.e. u that satisfy |u| = 1) if and only if (0, 1) u K ⊆ (0, 1) K for all unit length scalars u, in which case (0, 1) u K = (0, 1) K for all such u and p_K(s x) = |s| p_K(x) for all x ∈ X and all non-zero scalars s ≠ 0. If in addition p_K is also real-valued then this holds for all scalars (i.e. p_K is absolutely homogeneous).
   • Recall that p_K is called absolutely homogeneous if |s| p_K(x) is well-defined and p_K(s x) = |s| p_K(x) for all x ∈ X and all scalars s (not just non-zero scalars).
   • s K ⊆ K for all unit scalars s if and only if s K = K for all unit scalars s, in which case (0, 1) K = (0, 1) s K for all unit scalars s.

Proof —

Most of this theorem's statements follow immediately from the results established in the basic properties section below.

One statement that hasn't yet been proven, and which we now prove, is that a convex subset A of X that satisfies (0, ∞)A = X is absorbing in X. Suppose that X is a vector space over the field 𝔽, where 𝔽 is ℝ or ℂ, and assume that dim X ≠ 0. Observe that a set A is absorbing in X if and only if A ∩ span { x } is absorbing in every 1-dimensional vector subspace Y := span { x } = 𝔽 x, where x ≠ 0. Thus, it is necessary and sufficient to show that A ∩ Y contains an open 𝔽-ball around the origin in Y = 𝔽 x. The condition (0, ∞)A = X implies that every "open ray" in Y starting at the origin (i.e. a set of the form (0, ∞) y for some 0 ≠ y ∈ Y) contains an element of A so that in particular, A ∩ (0, ∞) y ≠ ∅ and A ∩ (0, ∞) (- y) ≠ ∅ (where (0, ∞) (- y) = (- ∞, 0) y so that A ∩ (- ∞, 0) y ≠ ∅) so that now the convexity of A ∩ Y makes it clear that for every 0 ≠ y ∈ Y, the convex set A ∩ ℝ y is a line segment (possibly open, closed, or half-closed, and possibly bounded or unbounded) containing an open sub-interval that contains the origin. If 𝔽 = ℝ then this shows that A ∩ Y is absorbing in Y = ℝ x so assume that 𝔽 = ℂ. Note that i x ∈ Y = ℂ x, where i = √-1, and that Y = ℂ x = (ℝ x) + (ℝ i x). Notice that the set (A ∩ ℝ x) ∪ (A ∩ ℝ (i x)), which is contained in A ∩ Y, is a union of two line segments intersecting at the origin (with each containing the origin in an open sub-interval) so that its convex hull, which is contained in the convex set A ∩ Y, clearly contains a quadrilateral having the origin in its interior. This shows that A ∩ Y is absorbing in Y, as desired.

Recall that the hypothesis of statement (6) allows us conclude that p_K(sx) = p_K(x) for all x ∈ X and all scalars s satisfying |s| = 1. Since every scalar s is of the form r e^{i t} for some real t where r := |s| ≥ 0 and e^{i t} is real if and only if s is real, the results stated in (6) follow immediately from the aforementioned conclusion, from the strict positive homogeneity of p_K, and from the positive homogeneity of p_K when p_K is real-valued.

Characterization of seminorms with Minkowski functionals

Note that in this next theorem, which follows immediately from the above corollary, we do not assume that K is absorbing in X but instead deduce that (0, 1) K is balanced, and that we do not assume that K is balanced (which is a property that K is often required to have) but we instead assume the weaker condition that (0, 1) s K ⊆ (0, 1) K for all scalars s satisfying |s| = 1. The common requirement that K be convex is also weakened to only requiring that (0, 1) K be convex

Theorem — Let K be a subset of a real or complex vector space X. Then p_K is a seminorm on X if and only if the following conditions hold:

1. (0, ∞) K = X (or equivalently, p_K is real-valued);
2. (0, 1) K is convex;
   • It suffices (but is not necessary) for K to be convex.
3. (0, 1) u K ⊆ (0, 1) K for all unit scalars u.
   • This condition is satisfied, in particular, if K is balanced or more generally if u K ⊆ K for all unit scalars u (i.e. scalars satisfying |u| = 1).

in which case 0 ∈ K and both (0, 1) K = { x ∈ X : p_K (x) < 1 } and ∩ε > 0 (0, 1 + ε) K = { x ∈ X : p_K (x) ≤ 1 } are convex, balanced, and absorbing in X.

Conversely, if f is a seminorm on X then the set V := { x ∈ X : f (x) < 1 } satisfies all of the above conditions, and thus also conclusions, and f = p_V; moreover, V is also convex, balanced, absorbing, and satisfies (0, 1) V = V = [0, 1] V.

Corollary — If K is a convex, balanced, and absorbing subset of a real or complex vector space X, then p_K is a seminorm on X.

Characterizing Minkowski functionals

The next theorem shows that Minkowski functionals are exactly those functions f : X → [0, ∞] that have a certain purely algebraic property that is used widely. This theorem can be easily extended to characterize certain classes of [-∞, ∞]-valued maps (e.g. real-valued sublinear functions) in terms of Minkowski functionals. In particular, we describe later how every real homogeneous function f : X → ℝ (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

Theorem — Let f : X → [0, ∞] be any function. The following are equivalent:

1. f is a Minkowski functional (i.e. there exists some subset K of X such that f = p_K).
2. Strict positive homogeneity: f (tx) = t f (x) for all x ∈ X and all positive real t > 0.
3. f = p_K where K := { x ∈ X : f(x) ≤ 1 }.
4. f = p_L where L := { x ∈ X : f(x) < 1 }.

Moreover, if f never takes on the value ∞ (so that the product 0 ⋅ f(x) is always well-defined) then we may add to this list:

5. Positive homogeneity/Nonnegative homogeneity: f (tx) = t f (x) for all x ∈ X and all non-negative real t ≥ 0.

Proof —

We prove only that (2) implies (3) since afterwards the rest of the theorem follows immediately from the basic properties of Minkowski functionals described later; properties that we will assume the reader is familiar with and use without mention. So assume that f : X → [0, ∞] is a function such that f (tx) = t f (x) for all x ∈ X and all real t > 0 and let K := { y ∈ X : f (y) ≤ 1 }.

Note that for all real t > 0 we have f (0) = f (t0) = t f (0) so by taking t = 2 for instance, we can conclude that either f (0) = 0 or f (0) = ∞. Let x ∈ X. We must show that f (x) = p_K(x).

We will now show that if f (x) = 0 or f (x) = ∞ then f (x) = p_K(x), so that in particular, we will conclude that f (0) = p_K(0). So suppose that f (x) = 0 or f (x) = ∞ and note that in either case we have f (tx) = t f (x) = f (x) for all real t > 0. Now if f (x) = 0 then this implies that that t x ∈ K for all real t > 0 (since f (tx) = 0 ≤ 1), which implies that p_K(x) = 0, as desired. Similarly, if f (x) = ∞ then t x ∉ K for all real t > 0, which implies that p_K(x) = ∞, as desired. Thus, we will henceforth assume that R := f (x) a positive real number and that x ≠ 0 (note, however, that we have not yet ruled out the possibility that p_K(x) is 0 or ∞).

Recall that just like f, p_K satisfies p_K(tx) = t p_K(x) for all real t > 0. Since 0 < 1/R < ∞, p_K(x) = R = f (x) if and only if p_K(1/R x) = 1 = f (1/R x) so we may assume without loss of generality that R = 1 and it remains to show that p_K(x) = 1. Since f (x) = 1, we have x ∈ K ⊆ (0, 1] K, which implies that p_K(x) ≤ 1 (so in particular, we now know that p_K(x) ≠ ∞). It remains to show that p_K(x) ≥ 1, which recall happens if and only if x ∉ (0, 1) K. So assume for the sake of contradiction that x ∈ (0, 1) K and let 0 < r < 1 and k ∈ K be such that x = r k, where note that k ∈ K implies that f (k) ≤ 1. Then 1 = f (x) = f (rk) = r f (k) ≤ r < 1. ∎

Minkowski functionals and negative values

The definitions of strict positive homogeneity that was given for [0, ∞]-valued functions on X immediately extends, without change, to functions that are valued in other codomains.

Definition: Let f be a function on X valued in [-∞, ∞] (or even in ℂ). We say that f is strictly positively homogeneous if f (tx) = t f (x) for all x ∈ X and all positive real t > 0. If f never takes the value ±∞ then we say that f is non-negative homogeneous if f (tx) = t f (x) for all x ∈ X and all non-negative real t ≥ 0.

Note that if g, h : X → [0, ∞] are functions such that g - h is well-defined (i.e. there is no x ∈ X such that g (x) and h (x) are infinite and equal), then if both g and h are strictly positively homogeneous (which happens if and only if they are both Minkowski functionals) then so is the map f := g - h : X → [-∞, ∞].

It's natural to ask if every function f : X → [-∞, ∞] that is strictly positively homogeneous is equal to the difference of two Minkowski functionals. The answer can easily be shown to be yes by using with the following definitions.

Positive and negative parts

Definition and notation: For any function f : X → [-∞, ∞], let f₊ : X → [0, ∞] and f_- : X → [0, ∞] be defined by f₊(x) := max { 0, f (x) } and f_-(x) := max { 0, -f (x) } = - min { 0, f (x) }. We call f₊ the positive part of f and call f_- the negative part of f.

For every x ∈ X, at least one of f₊ and f_- is equal to 0 (or said differently, the function min{ f₊, f_- } is identically 0) and f = f₊ - f_-. Moreover, f₊ and f_- are the only [0, ∞]-valued functions with these properties. That is, if g, h : X → [0, ∞] satisfy 0 = min { g, h } and f = g - h then f₊ = g and f_- = h.

Characterization

Theorem — Let f : X → [-∞, ∞] be a function. Then the following are equivalent:

1. f is strictly positively homogeneous.
2. f₊ and f_- are strictly positively homogeneous (or equivalently, they are Minkowski functionals).
3. f is equal to the (well-defined) difference of two Minkowski functionals.

Moreover, if f is real-valued then we may replace the terms "strictly positively homogeneous" with "non-negatively homogeneous" in the above characterization.

Linear functionals and Minkowski functionals

Definition: Call a map f : X → ℝ real homogeneous if f (tx) = t f (x) for all x ∈ X and all real t.

Clearly, f is real homogeneous if and only if it is non-negative homogeneous and f (- x) = - f (x) for all x ∈ X.

Definition: Call a map f : X → [-∞, ∞] additive if for all x, y ∈ X, f (x) + f (y) is well-defined and f (x + y) = f (x) + f (y).

If a map f : X → [-∞, ∞] is additive then either f is constant and infinite or else it is real-valued, so it is only useful to consider real-valued additive maps.

Definition: A real linear functional on X is a map f : X → ℝ that is additive and real homogeneous.

If f : X → ℝ is additive then f (0) = 0 and for every x ∈ X, f (tx) = t f (x) for all rational real t. Thus an additive function is almost real homogeneous and it is a linear functional if and only if it is also strictly positively homogeneous. Thus, we could have equivalently defined a real linear functional as being an additive map that is equal to the difference of two real-valued Minkowski functionals.

Commuting with negation

Notation: Let η : X → X denote negation on X (i.e. η(x) := - x for every x ∈ X).

Definition: We say that a function f on X commutes with negation if - f = f ∘ η. That is, if - f (x) = f (- x) for all x ∈ X.

Every ℝ-valued additive function commutes with negation, as does every real homogeneous function. The following observations shows that a function f : X → [-∞, ∞] that commutes with negation is completely determined by negation and the single [0, ∞]-valued map f₊ (the same is also true of f_-).

Observations — Let f : X → [-∞, ∞] be a function. Clearly, f (0) = - f (0) if and only if f (0) = 0. If x ∈ X, then - f (x) = f (- x) if and only if f₊ (x) = f_- (- x) and f_- (x) = f₊ (- x).

The following statements are equivalent:

1. f commutes with negation (i.e. - f = f ∘ η).
2. f_- = f₊ ∘ η.
3. f₊ = f_- ∘ η.
4. f = f₊  -  (f₊ ∘ η).
5. f = (f_- ∘ η)  -  f_-.

Minkowski functionals and real homogeneous functions

The following theorem shows that there is a one-to-one correspondence, determined by the negation operation η, between (ℝ-valued) real homogeneous functions and ℝ-valued Minkowski functionals g that satisfy min{ g, g ∘ η } = 0 (that is, min{ g(x), g(- x) } = 0 for all x ∈ X).

Theorem — If f : X → ℝ is real homogeneous (such as a real linear functional) then f₊ : X → [0, ∞) is a Minkowski functional satisfying 0 = min { f₊,  f₊ ∘ η } and f = f₊ - f₊ ∘ η (which implies f_- = f₊ ∘ η).

Conversely, every ℝ-valued Minkowski functional g : X → [0, ∞) satisfying 0 = min{ g, g ∘ η } determines a real homogeneous function f : X → ℝ defined by f := g - g ∘ η, where observe that f₊ = g and f_- = g ∘ η.

Basic properties

Summary of properties of Minkowski functionals

We summarize some of the more important results found in the following theorem where throughout K and L are any subsets of X and x ∈ X is arbitrary.

• p_K(tx) = t p_K(x) for all real t > 0 always, without any requirements at all.
  • If p_K is real-valued, which happens if and only if (0, ∞) K = X, then the above equality will also hold with t = 0.
• If 0 ≤ r ≤ ∞ then p_K(x) < r if and only if x ∈ (0, r) K.
  • The proofs of the many properties listed below become straightforward exercises once this characterization is proven and then the remaining properties follow easily once it is shown that every Minkowski functional is strictly positive homogeneous.
  • Assuming that 0 < R < ∞, if x ∈ (0, R] K then p_K(x) ≤ R, and the converse will hold if K contains { y ∈ X : p_K(y) = 1 }.
• p_L = p_K if and only if (0, 1)L = (0, 1)K.
  • It follows that p_L = p_K whenever L satisfies (0, 1) K ⊆ L ⊆ ∩ε > 0 (0, 1 + ε) K; in particular, L could be the first or last of these three sets or also (0, 1] K = K ∪ (0, 1) K.
• p_K is subadditive (i.e. satisfies the triangle inequality) if and only if (0, 1) K is convex. If K is convex then so are (0, 1) K and (0, 1] K = K ∪ (0, 1) K, and p_K is subadditive.
• If we let D := { y ∈ X : p_K(y) = 1 or p_K(y) = 0 } then p_D = p_K and D has the particularly nice property that for any positive real R > 0, x ∈ R D if and only if p_D(x) = R or p_D(x) = 0.
  • Moreover, for any positive real R > 0, p_D(x) ≤ R if and only if x ∈ (0, R] D.
  • If p_K(0) = 0 and if there is some point at which p_K takes on a positive real value, then D is necessarily not convex, balanced, or absorbing. However, the set (0, 1]D (or (0, 1)D = (0, 1)K, or others) may have some or all of these properties and since p_K = p_D = p_{(0, 1]D} one may use any of these sets to try to deduce properties of this map. That is, one is always free to use whatever set L is best suited for the problem at hand (so as long L satisfies p_L = p_K, of course) and then change to any other such set.
• The value(s) of p_K on a set of the form Y := (0, ∞) x (where x ∈ X) are completely determined by the value of p_K at any one point of Y. In general, the values of p_K on Y and its values on the complement X ∖ Y are completely independent.
  • This is the reason why characterizations such as: x ∉ (0, R) K if and only if K ∩ (1/R, ∞)x = ∅ are useful. The latter condition requires no change to K and involves only the open ray (0, ∞) x, which is all that the value of p_K(x) depends on, while the former condition involves scaling the entire set K.

Properties of Minkowski functionals

Theorem — Let K be any subset of a real or complex vector space X and let R ≥ 0 be a real number. Then for all x ∈ X:

1. Value at 0: p_K(0) ≠ ∞ if and only if 0 ∈ K if and only if p_K(0) = 0.
2. Infinite/Real values: p_K(x) = ∞ if and only if x ∉ (0, ∞) K if and only if K ∩ (0, ∞)x = ∅. Equivalently, p_K(x) ≠ ∞ if and only if x ∈ (0, ∞) K. It follows that:
   • (0, ∞) K is the set of all points on which p_K is real valued. In particular,
     • p_K is real-valued if and only if (0, ∞) K = X, in which case 0 ∈ K.
     • p_K is identically equal to ∞ if and only if K = ∅.
3. Null space: The following are equivalent:
   1. p_K(x) = 0.
   2. There exists a divergent sequence of positive real numbers (t_n)^∞
      _n=1 → ∞ such that t_n x ∈ K for all n.
      • In particular, if t x ∈ K for all real t > 0 then p_K(x) = 0.
   3. (0, ∞)x ⊆ (0, 1) K.
   4. x ∈ ∩ε > 0 (0, ε)K (that is, x ∈ (0, ε)K for all ε > 0).

   So ker p_K := { y ∈ X : p_K(y) = 0 } = ∩ε > 0 (0, ε) K and p_K is identically equal to 0 if and only if (0, 1) K = X.

4. p_K(x) < R if and only if x ∈ (0, R) K if and only if K ∩ (1/R, ∞)x ≠ ∅. Equivalently, p_K(x) ≥ R if and only if x ∉ (0, R) K if and only if K ∩ (1/R, ∞)x = ∅. It follows that:
   • { y ∈ X : p_K(y) < R } = (0, R) K.
   • This set equality and the two equivalences remain true if R is replaced by ∞ (this is the content of statement (2)).
     • Here, we take 1/∞ := 0 and 1/0 := ∞ so that (1/0, ∞) := { t ∈ ℝ : ∞ < t < ∞ } = ∅.
5. p_K(x) ≤ R if and only if x ∈ (0, R + ε) K for every ε > 0. It follows that:
   • { y ∈ X : p_K(y) ≤ R } = ∩ε > 0 (0, R + ε) K
     • Note that ∩ε > 0 (0, R + ε) K means ∩ε > 0 ((0, R + ε) K) and not (∩ε > 0 (0, R + ε)) K = (0, R] K.
   • (0, R] K ⊆ ∩ε > 0 (0, R + ε) K and if R > 0 then these sets are equal if and only if K contains { y ∈ X : p_K(y) = 1 }.
     • Thus, if x ∈ R K or x ∈ (0, R] K then p_K(x) ≤ R (but the converse is not necessarily true).
6. p_K(x) = R if and only if x ∉ (0, R) K and x ∈ (0, R + ε)K for every ε > 0. It follows that:
   • If x ∈ R K and x ∉ (0, R) K then p_K(x) = R (but the converse is not necessarily true).
   • p_K(x) = 1 if and only if x ≠ 0, K ∩ (1, ∞)x = ∅, and K ∩ (ε, 1]x ≠ ∅ for all ε < 1.
     • So if R > 0 then p_K(x) = R if and only if x ≠ 0, K ∩ (1/R, ∞)x = ∅, and K ∩ (ε, 1/R]x ≠ ∅ for all ε < 1/R.
     • One may restrict ε to range over any non-degenerate interval of the form (a, 1/R) or [a, 1/R).
7. Strict positive homogeneity: p_K(tx) = t p_K(x) for all real t > 0. It follows that:
   • 0 ⋅ p_K(x) is well-defined if and only if p_K(x) ≠ ∞ (or equivalently, x ∈ (0, ∞) K), in which case p_K(tx) = t p_K(x) for t = 0 if and only if 0 ∈ K.
   • Positive/Nonnegative homogeneity: p_K is nonnegative homogeneous if and only if p_K is real-valued.
     • Definition: p_K being nonnegative homogeneous on a set S means that r p_K(y) is well-defined and p_K(ry) = r p_K(y) for all y ∈ S and all non-negative real r ≥ 0. If S is not mentioned then it is understood that S := X.
   • If 0 ∈ K then (0, ∞) K is the unique maximal set of points in X on which p_K is nonnegative homogeneous. If 0 ∉ K then this set is empty.
8. Gauge equality: Let L be a subset of X. The following are equivalent:
   1. p_L = p_K.
   2. (0, 1) L = (0, 1) K.
   3. (0, r) L = (0, r) K for some (or equivalently, for all) real r > 0.
   4. ∩ε > 0 (0, 1 + ε) L = ∩ε > 0 (0, 1 + ε) K.
   5. ∩ε > 0 (0, r + ε) L = ∩ε > 0 (0, r + ε) K for some (or equivalently, for all) real r > 0.

   It follows that:

   • If (0, 1) K ⊆ L ⊆ ∩ε > 0 (0, 1 + ε) K then p_L = p_K. However, the converse is in general false.
     • In particular, if (0, 1) K ⊆ L ⊆ (0, 1] K then p_L = p_K.
   • If L is not a subset of ∩ε > 0 (0, 1 + ε) K then p_L ≠ p_K.
9. Gauge comparison: Let L be a subset of X. Then p_K ≤ p_L if and only if (0, 1) L ⊆ (0, 1) K.
10. Kernel ∪ Unit Sphere: Let D := { y ∈ X : p_K(y) = 1 or p_K(y) = 0 } = ∩r > 0 (0, r)K ∪ [( ∩ε > 0 (0, 1 + ε) K) ∖ (0, 1) K ].
    • p_D = p_K.
      • If p_K is 0 at any point of X, then this equality would not hold if D was instead defined to only be the unit sphere S := { y ∈ X : p_K(y) = 1 }, since we'd have p_S(x) = ∞ at any x where p_K(x) = 0.
    • If R > 0 then x ∈ R D if and only if p_D(x) = R or p_D(x) = 0.
    • K ⊆ (0, 1] K ⊆ (0, 1] D and if R > 0 then (0, R] D = { y ∈ X : p_D(y) ≤ R }.
    • (0, 1) K ⊆ D if and only if the range of p_K is contained in { 0, ∞ }.
11. Restriction: Let S be a real or complex vector subspace of X and let p_{K ∩ S} : S → [0, ∞] denote the Minkowski functional of K ∩ S on S. Then p_K|_S = p_{K ∩ S}, where p_K|_S denotes the restriction of p_K to S.
12. Subadditive/Triangle inequality: The following are equivalent:
    1. p_K satisfies the triangle inequality on X.
       • Definition: Recall that p_K is subadditive and satisfies the triangle inequality on a subset S of X if p_K(y + z) ≤ p_K(y) + p_K(z) for all y, z ∈ S. If mention of S is omitted then it is assumed that S := X.
    2. (0, 1) K is convex.
    3. ∩ε > 0 (0, 1 + ε) K is convex.
    • If K is convex then p_K is subadditive and (0, 1] K = K ∪ (0, 1) K is also convex.
    • If p_K is real-valued, then p_K is subadditive if and only if for all non-zero y, z ∈ X such that ℝ y ≠ ℝ z and p_K(x + y) = 1, we have 1 ≤ p_K(x) + p_K(y).
    • p_K(x + y) ≤ p_K(x) + p_K(y) whenever any of the following conditions are satisfied: y ∈ (∞, -1)x ∪ (-1, ∞)x, x = 0, y = 0, p_K(x) = ∞, p_K(y) = ∞, or p_K(x + y) = 0. If p_K(0) = 0 then it also holds when y = - x.
      • p_K is subadditive on any set of the form [0, ∞) x.
    • If p_K(0) = 0 (or if the real dimension of X is at least 2) then p_K is subadditive on X if and only if its restriction to every 2-dimensional real vector subspace of the form S = ℝy + ℝz is subadditive (where y, z ∈ X).
13. Invariance under scalar multiplication: If s ≠ 0 is a scalar then p_{s K}(y) = p_K(1/s y) for all y ∈ X. If r is real and 0 < r < ∞ then p_{r K}(x) = p_K(1/r x) = 1/r p_K(x). It follows that:
    • If s ≠ 0 is a scalar then the following are equivalent:
    1. p_K(s y) = p_K(y) for all y ∈ X.
    2. p_K(1/s y) = p_K(y) for all y ∈ X.
    3. p_K(sⁿ y) = p_K(y) for all y ∈ X and all integers n.
    4. p_{s K} = p_K.
    5. p_{1/s K} = p_K.
    6. p_{sⁿ K} = p_K for all integers n.
    7. (0, 1) s K = (0, 1) K.
    • Note that if s ≠ 1 is a positive real number then (0, 1) s K = (0, 1) K if and only if the range of p_K is contained in { 0, ∞ }.
    • p_K is symmetric (i.e. p_K(- y) = p_K(y) for all y ∈ X) if and only if p_K = p _{- K}, which happens if and only if (0, 1) K = - (0, 1) K.
      • Note that for any set S, - S ⊆ S if and only if - S = S, in which case (0, 1) S = - (0, 1) S.
    • p_K(u y) = p_K(y) for all y ∈ X and all unit length scalars u (i.e. u that satisfy |u| = 1) if and only if and only if p_K = p_{u K} for all unit length scalars u, which happens if and only if (0, 1) u K = (0, 1) K for all unit length scalars u.
      • Note that for any set S, u S ⊆ S for all unit length scalars u if and only if u S = S for all such u, in which case (0, 1) u S = (0, 1) S for all such u.
    • If s is a scalar such that sK = K, then p_K(s x) = p_K(x) if and only if s ≠ 0, K ≠ { 0 }, or x = 0 (see footnote for proof).[2]

Proof —

Most numbered statements in the theorem above have an unnumbered sub-list. We prove only the numbered statements and some of the less obvious statements found in the unnumbered sub-lists. All other statements are either direct consequences of the numbered statement under which it is found, or else will become straightforward exercises once all the numbered statements are proven.

Proof of (1) and (2): Straightforward exercises.

Proof of (3): The equivalence of (a) and (d) will follow immediately from (4) (or (5)), and the proofs of the remaining equivalences in (3) are straightforward exercises. ∎

Proof of (4): Let LHS (i.e. Left Hand Side) be the statement "p_K(x) < R" and let RHS be the statement "x ∈ (0, R) K". We must show that LHS is true if and only if RHS is true. If K = ∅ then LHS and RHS are both false so henceforth assume that K ≠ ∅. If R = 0 then LHS is clearly false and RHS is also false since (0, R) K = ∅ so henceforth assume that R > 0. If p_K(x) = ∞ then LHS is false (since R < ∞) and (2) gives x ∉ (0, ∞) K so that in particular, RHS is also false. So henceforth assume that p_K(x) ≠ ∞. Observe that LHS is true if and only if p_K(x) := inf { r ∈ ℝ : r > 0 and x ∈ rK } < R, which happens if and only if there exists some real r such that 0 < r < R and x ∈ rK, where this can be restated more succinctly as x ∈ (0, R) K, which is exactly the statement RHS. Thus LHS is true if and only if RHS is true, which completes the proof of (4). ∎

Proof of (5): Note that p_K(x) ≤ R if and only if for every p_K(x) < R + ε for every ε > 0, which by (4) happens if and only if x ∈ (0, R + ε)K for every ε > 0, as desired. The proof of the claim that if R > 0 then (0, R] K = ∩ε > 0 (0, R + ε) K if and only if this is true for R = 1 will be a straightforward exercise once (7) is proven. Then the claim that this equality holds for if and only if K contains { y ∈ X : p_K(y) = 1 } will be easily seen with the proof of (10). ∎

Proof of (6): The main characterization in (6) is just a combination of (4) and (5). The characterization of p_K(x) = 1 is now readily verified, and its generalization to p_K(x) = R for positive R will follow immediately once (7) is proven. ∎

Proof of (7): Fix 0 < t < ∞. If x = 0 then (1) implies that p_K(tx) = t p_K(x) so assume x ≠ 0. Note that p_K(x) = ∞ if and only if x ∉ (0, ∞) K if and only if tx ∉ (t0, t∞) K = (0, ∞) K if and only if p_K(tx) = ∞, in which case p_K(tx) = ∞ = t p_K(x) so we're done. So assume S := p_K(x) < ∞, which (as was just shown) implies that p_K(tx) < ∞ and K ≠ ∅. Since p_K(x) = S, by (6) we have x ∉ (0, S) K and x ∈ (0, S + ε)K for every ε > 0 so that multiplying by t > 0 gives tx ∉ (0, tS) K and tx ∈ (0, tS + tε)K for every ε > 0. Since ε > 0 was arbitrary, it follows that tx ∉ (0, tS) K and tx ∈ (0, tS + ε)K for every ε > 0, which is equivalent to p_K(tx) = tS, as desired. ∎

Proof of (8): The characterizations of when p_L = p_K are now deduced easily from the following observation: if f and g are any [0, ∞]-valued functions on X, then f = g if and only if { y ∈ X : f(y) < r } = { y ∈ X : g(y) < r } for all real 0 < r < ∞ (importantly, note that it is not necessary to check that these set equalities hold for r = 0 or r = ∞). Note that this characterization of f = g remains true if all instances of "< r" are replaced by "≤ r". Using (4) or (7), it is easy to see that p_L = p_K holds if and only if { y ∈ X : p_L(y) < r } = { y ∈ X : p_K(y) < r } for some real r > 0 (in which case this will be true for all positive reals). Similarly, (7) shows that this characterization of p_L = p_K remains true if "< r" is replaced by "≤ r". For the statement involving the set D, it is readily verified that (0, 1) D = (0, 1) K so that p_L = p_K, and the rest now follows with the help of (5). ∎

Proof of (9): Immediate.

Proof of (10): Recall that (0, 1)K = { y ∈ X : p_K(y) < 1 } so by using (7), we have

(0, 1)D = { ty ∈ X : 0 < t < 1, p_K(y) = 1 or p_K(y) = 0 } = { ty ∈ X : 0 < t < 1, p_K(ty) = t or p_K(ty) = 0 } = (0, 1)K,

so (8) implies that p_D = p_K. The rest of (10) follows easily. ∎

Proof of (11): Immediate.

Proof of (12): It is straight forward to show that if p_K is subadditive then (0, 1) K = { y ∈ X : p_K(y) < 1} and { y ∈ X : p_K(y) ≤ 1 } = ∩ε > 0 (0, R + ε) K are convex.

Assume that K is convex, which recall is true if and only if sK + tK = (s + t)K for all non-negative real s and t. We will show that p_K is subadditive. Let x, y ∈ X and note that if R := p_K(x) is infinite or S := p_K(y) is infinite then we're done, so assume that both are finite. We want to show that p_K(x + y) ≤ R + S, which is true if and only if x + y ∈ (0, R + S + ε)K for every ε > 0. Fix ε > 0. Since p_K(x) ≤ R, we have x ∈ (0, R + ε/2)K and similarly we have y ∈ (0, S + ε/2)K so that x + y ∈ (0, R + ε/2)K + (0, S + ε/2)K = (0, R + S + ε)K, where the last equality is due to K being convex. ∎

If K is convex then p_K is subadditive, which implies that (0, 1) K is convex, where this fact can now be used to easily prove that (0, 1] K = K ∪ (0, 1) K is also convex.

If y, z ∈ X are such that p_K(y + z) > p_K(xy) + p_K(z), then p_K is not subadditive on ℝy + ℝz, so it suffices to check subadditivity on sets of the form ℝy + ℝz.

If y ∈ (0, ∞)x then p_K(x + y) ≤ p_K(x) + p_K(y) is immediate from p_K being strictly positively homogeneous, while if y = - tx = t(- x) with 1 ≠ t > 0 then this inequality follows from the appropriate choice of one of the equalities: x + y = (1 - t)x = (t - 1)(- x)). That p_K is subadditive on any set of the form [0, ∞) x is now immediate. If Q := p_K(x + y) > 0 is real (we assume that Q > 0 since if Q = 0 then we're done) then p_K(x + y) = Q ≤ p_K(x) + p_K(y) if and only if p_K(1/Qx + 1/Qy) = 1 ≤ p_K(1/Qx) + p_K(1/Qy) so in this case, it suffices to prove the triangle inequality under the assumption that p_K(x + y) = 1. ∎

Proof of (13): The proofs are straightforward exercises that follow easily from the above characterizations. ∎

Examples

1. If ℒ is a non-empty collection of subsets of X then p_∪ℒ (x) = inf { p_L(x) : L ∈ ℒ } for all x ∈ X, where ∪ℒ := ∪L ∈ ℒ L.
   • Thus p_K∪L(x) = min { p_K(x), p_L(x) } for all x ∈ X.
2. If ℒ is a non-empty collection of subsets of X and I ⊆ X satisfies

   { x ∈ X : p_L(x) < 1 for all L ∈ ℒ } ⊆ I ⊆ { x ∈ X : p_L(x) ≤ 1 for all L ∈ ℒ }

   then p_I (x) = sup { p_L(x) : L ∈ ℒ } for all x ∈ X.

We now give examples where the containment (0, R] K ⊆ ∩ε > 0 (0, R + ε) K is proper.

Example: Note that if R = 0 and K = X then (0, R] K = (0, 0] X = ∅ but ∩ε > 0 (0, ε) K = ∩ε > 0 X = X, which shows that its possible for (0, R] K to be a proper subset of ∩ε > 0 (0, R + ε) K when R = 0. ∎

We now show that the containment can be proper when R = 1 with an example that may be generalized to any real R > 0. Assuming that [0, 1] K ⊆ K, the following example is representative of how it happens that x ∈ X satisfies p_K(x) = 1 but x ∉ (0 ,1] K.

Example: Let x ∈ X be non-zero and let K := [0, 1) x where note that [0, 1] K = K and x ∉ K. Since x ∉ (0, 1) K = K we have p_K(x) ≥ 1. That p_K(x) ≤ 1 follows from observing that for every ε > 0 we have (0, 1 + ε) K = (0, 1 + ε)([0, 1) x) = [0, 1 + ε) x, which contains x. Thus p_K(x) = 1 and x ∈ ∩ε > 0 (0, 1 + ε) K. However, (0, 1] K = (0, 1]([0, 1) x) = [0, 1) x = K so that x ∉ (0, 1] K, as desired. ∎

Less general conditions making a gauge into a seminorm

So that we have p_K(0) = 0, we will henceforth assume that 0 ∈ K.

If K is absorbing in X then one may show that p_{[0, 1]K} is positive homogeneous i.e. that p_{[0, 1]K}(sx) = s p_{[0, 1]K}(x) for all real s ≥ 0, where [0, 1]K := { tk : 0 ≤ t ≤ 1, k ∈ K }.[3] If q is a non-negative real-valued function on X that is positive homogeneous then the sets U := { x ∈ X : q(x) < 1 } and D := { x ∈ X : q(x) ≤ 1 } satisfy [0, 1] U = U and [0, 1] D = D; if in addition q is absolutely homogeneous then both U and D are balanced.[3]

If K is convex and the origin to belong to the algebraic interior of K, then p_K is a non-negative sublinear functional on X, which implies in particular that it is subadditive and positive homogeneous.

In order for p_K to be a seminorm, it suffices for K to be a disk (i.e. convex and balanced) and absorbing in X, which are the most common assumption placed on K.

Theorem[4] — If K is an absorbing disk in a vector space X then the Minkowski functional of K, which is the map p_K : X → [0, ∞) defined by

p_K(x) := inf { r > 0 : x ∈ rK},

is a seminorm on X. Moreover,

p_K(x) = 1/sup { r > 0 : rx ∈ K } .

Gauges of absorbing disks

Arguably the most common requirements placed on a set K to guarantee that p_K is a seminorm are that K be an absorbing disk in X. Due to how common these assumptions are, we now investigate the properties of a Minkowski functional p_K when K is an absorbing disk. Since all of the results mentioned above made few (if any) assumptions on K, they can be applied in this special case.

Theorem — We assume that K is an absorbing subset of X. We show that:

1. If K is convex then p_K is subadditive.
2. if K is balanced then p_K is absolutely homogeneous (i.e. p_K(s x) = |s| p_K(x) for all scalars s).

Proving a that the Gauge of an absorbing disk is a seminorm

Proof —

Convexity and subadditivity

A simple geometric argument that shows convexity of K implies subadditivity is as follows. Suppose for the moment that p_K(x) = p_K(y) = r. For all ε > 0, we have x, y ∈ K_ε := (r + ε) K. Since K is convex and r + ε ≠ 0, K_ε is also convex. Therefore, ½ x + ½ y ∈ K_ε. By definition of the Minkowski functional p_K, we have

${\displaystyle p_{K}\left({\frac {1}{2}}x+{\frac {1}{2}}y\right)\leq r+\epsilon ={\frac {1}{2}}p_{K}(x)+{\frac {1}{2}}p_{K}(y)+\epsilon }$.

But the left hand side is ½ p_K(x + y) so that p_K(x + y) ≤ p_K(x) + p_K(y) + 2ε.

Since ε > 0 was arbitrary, it follows that p_K(x + y) ≤ p_K(x) + p_K(y), which is the desired inequality. The general case p_K(x) > p_K(y) is obtained after the obvious modification.

Note: Convexity of K, together with the initial assumption that the set {r > 0: x ∈ r K} is nonempty, implies that K is absorbing.

Balancedness and absolute homogeneity

Notice that K being balanced implies that

${\displaystyle \lambda x\in rK\quad {\mbox{if and only if}}\quad x\in {\frac {r}{|\lambda |}}K.}$

Therefore

${\displaystyle p_{K}(\lambda x)=\inf \left\{r>0:\lambda x\in rK\right\}=\inf \left\{r>0:x\in {\frac {r}{|\lambda |}}K\right\}=\inf \left\{|\lambda |{\frac {r}{|\lambda |}}>0:x\in {\frac {r}{|\lambda |}}K\right\}=|\lambda |p_{K}(x).}$

Algebraic properties

Let X be a real or complex vector space and let K be an absorbing disk in X.

• p_K is a seminorm on X.
• p_K is a norm on X if and only if K does not contain a non-trivial vector subspace.[5]
• For any scalar s ≠ 0, p_sK = 1/|s|p_K.[5]
• If J is an absorbing disk in X and J ⊆ K then p_K ≤ p_J
• If K is a set satisfying { x ∈ X : p(x) < 1 } ⊆ K ⊆ { x ∈ X : p(x) ≤ 1 } then K is absorbing in X and p = p_K, where p_K is the Minkowski functional associated with K (i.e. the guage of K).[6]
  • In particular, if K is as above and q is any seminorm on X, then q = p if and only if { x ∈ X : q(x) < 1 } ⊆ K ⊆ { x ∈ X : q(x) ≤ 1 }.[6]
• If x ∈ X satisfies p_K(x) < 1 then x ∈ K.

Topological properties

Let X be a real or complex topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let K be an absorbing disk in X.

• Int K   ⊆   { x ∈ X : p_K(x) < 1 }   ⊆   K   ⊆   { x ∈ X : p_K(x) ≤ 1 }   ⊆   Cl K

  where Int K is the topological interior and Cl K is the closure of K in X.[7]

  • Note that we did not assume that p_K was continuous or that K had any topological properties.
• p_K is continuous if and only if K is a neighborhood of the origin in X.[7]
• If p_K is continuous then:[7]
  • Int K = { x ∈ X : p_K(x) < 1}, and
  • Cl K = { x ∈ X : p_K(x) ≤ 1}.

Motivating examples

Example 1

Consider a normed vector space X, with the norm ||·||. Let K be the unit ball in X. Define a function ${\displaystyle p:X\rightarrow \mathbb {R} }$ by

${\displaystyle p(x)=\inf \left\{r>0:x\in rK\right\}.}$

One can see that ${\displaystyle p(x)=\|x\|}$, i.e. p is just the norm on X. The function p is a special case of a Minkowski functional.

Example 2

Let X be a vector space without topology with underlying scalar field ${\displaystyle \mathbb {K} }$. Take ${\displaystyle \phi \in X'}$, the algebraic dual of X, i.e. ${\displaystyle \phi :X\rightarrow \mathbb {K} }$ is a linear functional on X. Fix a > 0. Let the set K be given by

${\displaystyle K=\{x\in X:|\phi (x)|\leq a\}.}$

Again we define

${\displaystyle p(x)=\inf \left\{r>0:x\in rK\right\}.}$

Then

${\displaystyle p(x)={\frac {1}{a}}|\phi (x)|.}$

The function p(x) is another instance of a Minkowski functional. It has the following properties:

1. It is subadditive: ${\displaystyle p(x+y)\leqslant p(x)+p(y)}$,
2. It is homogeneous: for all scalars s ${\displaystyle p(sx)=\left\vert s\right\vert p(x)}$,
3. It is nonnegative.

Therefore, p is a seminorm on X, with an induced topology. This is characteristic of Minkowski functionals defined via "nice" sets. There is a one-to-one correspondence between seminorms and the Minkowski functional given by such sets. What is meant precisely by "nice" is discussed in the section below.

Notice that, in contrast to a stronger requirement for a norm, p(x) = 0 need not imply x = 0. In the above example, one can take a nonzero x from the kernel of ${\displaystyle \phi }$. Consequently, the resulting topology need not be Hausdorff.

See also

• Hadwiger's theorem
• Hugo Hadwiger – Swiss mathematician
• Locally convex topological vector space – Type of topological vector space
• Morphological image processing
• Norm (mathematics) – Length in a vector space
• Seminorm
• Topological vector space – Vector space with a notion of continuity

References

1. Narici 2011, p. 109. sfn error: no target: CITEREFNarici2011 (help)
2. Proof: Let LHS be the statement "p_K(sx) = p_K(x)" and let RHS be the statement "s ≠ 0, K ≠ { 0 }, or x = 0". We want to show that LHS is true if and only if RHS is true. (⇐) Assume that RHS is true. Clearly, LHS is true if K = ∅ or if x = 0 so assume henceforth that K ≠ ∅ and x ≠ 0. Observe that if s = 0 then sK = K implies that K ⊆ { 0 }, where K ≠ ∅ now implies that K = { 0 }, but this contradicts the statement RHS, which we assumed was true. Thus we must have s ≠ 0. Note that p_K(x) = ∞ ⇔ x ∉ (0, ∞) K ⇔ sx ∉ (0, ∞) sK = (0, ∞) K (since s ≠ 0) ⇔ p_K(sx) = ∞ so we may now assume that both p_K(x) and p_K(sx) are finite. Let R := p_K(x). Then x ∉ (0, R) K and x ∈ (0, R + ε)K for every ε > 0 so that (since sx ≠ 0) sx ∉ (0, R) sK and sx ∈ (0, R + ε)sK for every ε > 0, where now using sK = K allows us to conclude that p_K(sx) = R, as desired. (⇒) Assume that RHS is false so that s = 0, K = { 0 }, and x ≠ 0. Since s = 0 and 0 ∈ K, we have 0 = p_K(sx). Since K = { 0 } and x ≠ 0, we have p_K(x) = ∞. Thus 0 = p_K(sx) ≠ p_K(x) = ∞, which shows that LHS is false, as desired. ∎
3. Jarchow 1981, pp. 104-108. sfn error: no target: CITEREFJarchow1981 (help)
4. Narici 2011, p. 119. sfn error: no target: CITEREFNarici2011 (help)
5. Narici 2011, pp. 115-154. sfn error: no target: CITEREFNarici2011 (help)
6. Schaefer 1999, p. 40.
7. Narici 2011, p. 119-120. sfn error: no target: CITEREFNarici2011 (help)

• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.CS1 maint: ref=harv (link)
• Diestel, Joe (July 30, 2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. 16. Providence, R.I: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Dineen, Seán (1981). Complex Analysis in Locally Convex Spaces. North-Holland Mathematics Studies. 57. Amsterdam New York New York: North-Holland Pub. Co., Elsevier Science Pub. Co. ISBN 978-0-08-087168-4. OCLC 16549589.CS1 maint: ref=harv (link)
• Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. Pure and applied mathematics. 1. New York: Wiley-Interscience. ISBN 978-0-471-60848-6. OCLC 18412261.
• Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Hogbe-Nlend, Henri (January 15, 1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. OCLC 316549583.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Hogbe-Nlend, Henri; Moscatelli, V. B. (January 1, 1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. 52. Amsterdam New York New York: North Holland. ISBN 978-0-08-087163-9. OCLC 316564345.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Husain, Taqdir; Khaleelulla, S. M. (December 5, 1978). Written at Berlin Heidelberg. Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin New York: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
• Keller, Hans (November 15, 1974). Differential Calculus in Locally Convex Spaces. Lecture Notes in Mathematics. 417. Berlin New York: Springer-Verlag. ISBN 978-3-540-06962-1. OCLC 1103033.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
• Köthe, Gottfried (December 19, 1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Pietsch, Albrecht (July 1, 1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Robertson, Alex P.; Robertson, Wendy J. (January 1, 1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Thompson, Anthony C. (1996). Minkowski Geometry. Encyclopedia of Mathematics and Its Applications. Cambridge University Press. ISBN 0-521-40472-X.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Schechter, Eric (October 30, 1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.CS1 maint: date and year (link)
• Schaefer, H. H. (1999). Topological Vector Spaces. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.CS1 maint: ref=harv (link)
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
• Wong, Yau-Chuen (July 1, 1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.CS1 maint: ref=harv (link) CS1 maint: date and year (link)