Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector spaces (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric).

Pseudometrics and metrics

A pseudometric on a set X is a map d : X × X → ℝ satisfying the following properties:

  1. d(x, x) = 0 for all xX;
  2. Symmetry: d(x, y) = d(y, x) for all x, yX;
  3. Subadditivity: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, zX.

A pseudometric is called a metric if it satisfies:

  1. Identity of indiscernibles: for all x, yX, if d(x, y) = 0 then x = y.
Ultrapseudometric

A pseudometric d on X is called a ultrapseudometric or a strong pseudometric if it satisfies:

  1. Strong/Ultrametric triangle inequality: for all x, yX, d(x, y) ≤ max { d(x, z), d(y, z)}.
Pseudometric space
Definition: A pseudometric space is a pair (X, d) consisting of a set X and a pseudometric d on X such that X's topology is identical to the topology on X induced by d. We call a pseudometric space (X, d) a metric space (resp. ultrapseudometric space) when d is a metric (resp. ultrapseudometric).

Topology induced by a pseudometric

If d is a pseudometric on a set X then collection of open balls:

Br(z) := { xX : d(x, z) < r}, as z ranges over X and r ranges over the positive real numbers,

forms a basis for a topology on X that is called the d-topology or the pseudometric topology on X induced by d.

Convention: If (X, d) is a pseudometric space and X is treated as a topological space, then unless indicated otherwise, it should be assumed that X is endowed with the topology induced by d.
Pseudometrizable space
Definition: A topological space (X, τ) is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d on X such that τ is equal to the topology induced by d.[1]

Pseudometrics and values on topological groups

Definition: An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
Definition: A topology τ on a real or complex vector space X is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (i.e. if it makes X into a topological vector space).

Every topological vector space (TVS) X is an additive commutative topological group but not all group topologies on X are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space X may fail to make scalar multiplication continuous. For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

Translation invariant pseudometrics

If X is an additive group then we say that a pseudometric d on X is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

  1. Translation invariance: d(x + z, y + z) = d(x, y) for all x, y, zX;
  2. d(x, y) = d(x - y, 0) for all x, yX.

Value/G-seminorm

If X is a topological group the a value or G-seminorm on X (the G stands for Group) is a real-valued map p : X → ℝ with the following properties:[2]

  1. Non-negative: p ≥ 0.
  2. Subadditive: p(x+y) ≤ p(x) + p(y) for all x, yX;
  3. p(0) = 0.
  4. Symmetric: p(-x) = p(x) for all xX.

where we call a G-seminorm a G-norm if it satisfies the additional condition:

  1. Total/Positive definite: If p(x) = 0 then x = 0.

Properties of values

If p is a value on a vector space X then:

  • |p(x) - p(y)|p(x - y) for all x, yX.[3]
  • p(nx) ≤ np(x) and 1/np(x) ≤ p(x/n)for all xX and positive integers n.[4]
  • The set { xX : p(x) = 0} is an additive subgroup of X.[3]

Equivalence on topological groups

Theorem[2]  Suppose that X is an additive commutative group. If d is a translation invariant pseudometric on X then the map p(x) := d(x, 0) is a value on X called the value associated with d, and moreover, d generates a group topology on X (i.e. the d-topology on X makes X into a topological group). Convergesly, if p is a value on X then the map {{{1}}} is a translation-invariant pseudometric on X and the value associated with d is just p.

Pseudometrizable topological groups

Theorem[2]  If (X, τ) is an additive commutative topological group then the following are equivalent:

  1. τ is induced by a pseudometric; (i.e. (X, τ) is pseudometrizable);
  2. τ is induced by a translation-invariant pseudometric;
  3. the identity element in (X, τ) has a countable neighborhood basis.

If (X, τ) is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric." A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable.

An invariant pseudometric that doesn't induce a vector topology

Let X be a non-trivial (i.e. X ≠ { 0}) real or complex vector space and let d be the translation-invariant trivial metric on X defined by d(x, x) = 0 and d(x, y) = 1 for all x, yX such that xy. The topology τ that d induces on X is the discrete topology, which makes (X, τ) into a commutative topological group under addition but does not form a vector topology on X because (X, τ) is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on (X, τ).

This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.

Paranorms

If X is a vector space over the real or complex numbers then a paranorm on X is a G-seminorm (defined above) p : X → ℝ on X that satisfies any of the following additional conditions, each of which begins with "for all sequences x = (xi)
i=1
in X and all convergent sequences of scalars s = (si)
i=1
":[5]

  1. Continuity of multiplication: if s is a scalar and xX are such that p(xi - x) → 0 and ss, then p(si xi - sx) → 0.
  2. Both of the conditions:
    • if s → 0 and if xX is such that p(xi - x) → 0 then p(si xi) → 0;
    • if p(xi) → 0 then p(s xi) → 0 for every scalar s.
  3. Both of the conditions:
    • if p(xi) → 0 and ss for some scalar s then p(si xi) → 0;
    • if s → 0 then p(si x) → 0 for all xX.
  4. Separate continuity:[6]
    • if ss for some scalar s then p(si x - sx) → 0 for every xX;
    • if s is a scalar, xX, and p(xi - x) → 0 then p(s xi - sx) → 0.

A paranorm is called an total if in addition it satisfies:

  • Total/Positive definite: p(x) = 0 implies x = 0.

Properties of paranorms

  • If p is a paranorm on a vector space X then the map d : X × X → ℝ defined by d(x, y) := p(x - y) is a translation-invariant pseudometric on X that defines a vector topology on X.[7]

If p is a paranorm on a vector space X then:

  • the set { xX : p(x) = 0} is a vector subspace of X.[7]
  • p(x + n) = p(x) for all x, nX with p(n) = 0.[7]
  • If a paranorm p satisfies p(sx) ≤ |s| p(x) for all xX and scalars s, then p is absolutely homogeneity (i.e. equality holds)[7] and thus p is a seminorm.

Examples of paranorms

  • If d is a translation-invariant pseudometric on a vector space X that induces a vector topology τ on X (i.e. (X, τ) is a TVS) then the map p(x) := d(x - y, 0) defines a continuous paranorm on (X, τ); moreover, the topology that this paranorm p defines in X is τ.[7]
  • If p is a paranorm on X then so is the map q(x) := p(x)/[1 + p(x)].[7]
  • Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
  • Every seminorm is a paranorm.[7]
  • The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).[8]
  • The sum of two paranorms is a paranorm.[7]
  • If p and q are paranorms on X then so is (pq)(x) := inf { p(y) + q(z) : x = y + z with y, zX}. Moreover, pqp and pqq. This makes the set of paranorms on X into a conditionally complete lattice.[7]
  • Each of the following real-valued maps are paranorms on X := ℝ2:
    • (x, y) ↦ |x|
    • (x, y) ↦ |x| + |y|
  • The real-valued map (x, y) ↦ x2 + y2 is not paranorms on X := ℝ2.[7]
  • If x = (xi)iI is a Hamel basis on a vector space X then the real-valued map that sends x = iI sixiX (where all but finitely many of the scalars si are 0) to iI |si| is a paranorm on X, which satisfies p(sx) = |s| p(x) for all xX and scalars s.[7]
  • The function p(x) := |sin(πx)| + min {2, |x|} is a paranorm on that is not balanced but nevertheless equivalent to the usual norm on R. Note that the function x|sin(πx)| is subadditive.[9]
  • Let X be a complex vector space and let X denote X considered as a vctor space over . Any paranorm on X is also a paranorm on X.[8]

F-seminorms

If X is a vector space over the real or complex numbers then an F-seminorm on X (the F stands for Fréchet) is a real-valued map p : X → ℝ with the following properties:[10]

  1. Non-negative: p ≥ 0.
  2. Subadditive: p(x+y) ≤ p(x) + p(y) for all x, yX;
  3. Balanced: p(ax) ≤ p(x) for all xX and all scalars a satisfying |a| ≤ 1 ;
    • This condition guarantees that each set of the form { xX : p(x) ≤ r } or { xX : p(x) < r } for some r ≥ 0 is balanced.
  4. for every xX, p(1/n x) → 0 as n → ∞
    • The sequence (1/n)
      n=1
      can be replaced by any positive sequence converging to 0.[11]

An F-seminorm is called an F-norm if in addition it satisfies:

  1. Total/Positive definite: p(x) = 0 implies x = 0.

An F-seminorm is called monotone if it satisfies:

  1. Monotone: p(rx) < p(sx) for all non-zero xX and all real s and t such that s < t.[11]

F-seminormed spaces

Definition:[11] An F-seminormed space (resp. F-normed space) is a pair (X, p) consisting of a vector space X and an F-seminorm (resp. F-norm) p on X.
Definition:[11] If (X, p) and (Z, q) are F-seminormed spaces then a map f : XZ is called an isometric embedding if q(f (x) - f (y)) = p(x - y) for all x, yX.

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.[11]

Examples of F-seminorms

  • Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
  • The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
  • If p and q are F-seminorms on X then so is their pointwise supremum x ↦ sup { p(x), q(x)}. The same is true of the supremum of any non-empty finite family of F-seminorms on X.[11]
  • The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).[8]
  • A non-negative real-valued function on X is a seminorm if and only if it is a convex F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm.[9]
    • In particular, every seminorm is an F-seminorm.
  • For any 0 < p < 1, the map f on n defined by [f(x1, ..., xn)]p := |x1|p + ⋅⋅⋅ + |xn|p is an F-norm that is not a norm.
  • If L : XY is a linear map and if q is an F-seminorm on Y, then qL is an F-seminorm on X.[11]
  • Let X be a complex vector space and let X denote X considered as a vctor space over . Any F-seminorm on X is also an F-seminorm on X.[8]

Properties of F-seminorms

  • Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[12]
  • Every F-seminorm on a vector space X is a value on X. In particular,
    • p(0) = 0;
    • p(x) = p(-x) for all xX.

Topology induced by a single F-seminorm

Theorem[10]  Let p be an F-seminorm on a vector space X. Then the map d : X × X → ℝ defined by d(x, y) := p(x - y) is a translation invariant pseudometric on X that defines a vector topology τ on X. If p is an F-norm then d is a metric. When X is endowed with this topology then p is a continuous map on X.

The balanced sets { x X : p(x) ≤ r}, as r ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets { x X : p(x) < r}, as r ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.

Topology induced by a family of F-seminorms

Suppose that 𝒮 is a non-empty collection of F-seminorms on a vector space X and for any finite subset ℱ ⊆ 𝒮 and any r > 0, let

Uℱ, r := p ∈ ℱ { xX : p(x) < r}.

The set { Uℱ, r : r > 0, ℱ ⊆ 𝒮, ℱ finite} forms a filter base on X that also forms a neighborhood basis at the origin for a vector topology on X denoted by τ𝒮.[11]

  • Each Uℱ, r is a balanced and absorbing subset of X.[11]
  • Uℱ, r/2 + Uℱ, r/2Uℱ, r.[11]
  • τ𝒮 is the coarsest vector topology on X making each p ∈ 𝒮 continuous.[11]
  • τ𝒮 is Hausdorff if and only if for every non-zero xX, there exists some p ∈ 𝒮 such that p(x) > 0.[11]
  • If 𝒯 is the set of all continuous F-seminorms on (X, τ𝒮) then τ𝒮 = τ𝒯.[11]
  • If 𝒯 is the set of all pointwise suprema of non-empty finite subsets of of 𝒮 then 𝒯 is a directed family of F-seminorms and τ𝒮 = τ𝒯.[11]

Fréchet combination

Suppose that p = (pi)
i=1
is a family of non-negative subadditive functions on a vector space X.

Definition:[7] The Fréchet combination of p is defined to be the real-valued map

.

As an F-seminorm

Assume that p = (pi)
i=1
is an increasing sequence of seminorms on X and let p be the Fréchet combination of p. Then p is an F-seminorm on X that induces the same locally convex topology as the family p of seminorms.[13]

Since p = (pi)
i=1
is increasing, a basis of open neighborhoods of the origin consists of all sets of the form { xX : pi(x) < r } as i ranges over all positive integers and r > 0 ranges over all positive real numbers.

The translation invariant pseudometric on X induced by this F-seminorm p is

(this metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations).[14]

As a paranorm

If each pi is a paranorm then so is p and moreover, p induces the same topology on X as the family p of paranorms.[7] This is also true of the following paranorms on X:

  • q(x) := inf { n
    i=1
    pi(x) + 1/n : n > 0 is an integer
    }.[7]
  • r(x) :=
    n=1
    min { 1/2n, pn(x)
    }.[7]

Generalization

The Fréchet combination can be generalized by use of a bounded remetrization function.

Definition:[15] A bounded remetrization function is a continuous non-negative non-decreasing map R : [0, ∞) → [0, ∞) that is subadditive (i.e. R (s + t) ≤ R (s) + R (t) for all s, t ≥ 0), has a bounded range, and satisfies R (s) = 0 if and only if s = 0.

Examples of bounded remetrization functions include arctan t, tanh t, t ↦ min { t, 1}, and tt/1 + t.[15] If d is a pseudometric (resp. metric) on X and R} is a bounded remetrization function then Rd is a bounded pseudometric (resp. bounded metric) on X that is uniformly equivalent to d.[15]

Suppose that p = (pi)
i=1
is a family of non-negative F-seminorm on a vector space X, R} is a bounded remetrization function, and r = (ri)
i=1
is a sequence of positive real numbers whose sum is finite. Then

defines a bounded F-seminorm that is uniformly equivalent to the p.[16] It has the property that for any net x = (xi)aA in X, p (x) → 0 if and only if pi (x) → 0 for all i.[16] p is an F-norm if and only if the p separate points on X.[16]

Characterizations

Of pseudometrizable TVS

If (X, τ) is a topological vector space (TVS) (where note in particular that τ is assumed to be a vector topology) then the following are equivalent:[10]

  1. X is pseudometrizable (i.e. the vector topology τ is induced by a pseudometric on X).
  2. X has a countable neighborhood base at the origin.
  3. The topology on X is induced by a translation-invariant pseudometric on X.
  4. The topology on X is induced by an F-seminorm.
  5. The topology on X is induced by a paranorm.

Of metrizable TVS

If (X, τ) is a TVS then the following are equivalent:

  1. X is metrizable.
  2. X is Hausdorff and pseudometrizable.
  3. X is Hausdorff and has a countable neighborhood base at the origin.[10][11]
  4. The topology on X is induced by a translation-invariant metric on X.[10]
  5. The topology on X is induced by an F-norm.[10][11]
  6. The topology on X is induced by a monotone F-norm.[11]
  7. The topology on X is induced by a total paranorm.

Of locally convex pseudometrizable TVS

If (X, τ) is TVS then the following are equivalent:[13]

  1. X is locally convex and pseudometrizable.
  2. X has a countable neighborhood base at the origin consisting of convex sets.
  3. The topology of X is induced by a countable family of (continuous) seminorms.
  4. The topology of X is induced by a countable increasing sequence of (continuous) seminorms (pi)
    i=1
    (increasing means that for all i, pipi+1).
  5. The topology of X is induced by an F-seminorm of the form:
    where (pi)
    i=1
    are (continuous) seminorms on X.[17]

Quotients

Let M be a vector subspace of a topological vector space (X, τ).

  • If X is a pseudometrizable TVS then so is X/M.[10]
  • If X is a complete pseudometrizable TVS and M is a closed vector subspace of X then X/M is complete.[10]
  • If X is metrizable TVS and M is a closed vector subspace of X then X/M is metrizable.[10]
  • If p is an F-seminorm on X, then the map P : X/M → ℝ defined by
    P(x + M) := inf { p(x + m) : mM}

    is an F-seminorm on X/M that induces the usual quotient topology on X/M.[10]

    • If in addition p is an F-norm on X and if M is a closed vector subspace of X then P is an F-norm on X.[10]

Examples and sufficient conditions

  • Every seminormed space (X, p) is pseudometrizable with a canonical pseudometric given by d(x, y) := p(x - y) for all x, yX.[18].
  • If (X, d) is pseudometric TVS with a translation invariant pseudometric d, then p(x) := d(x, 0) defines a paranorm.[19]
    • However, if d is a translation invariant pseudometric on the vector space X (without the addition condition that (X, d) is pseudometric TVS), then d need not be either an F-seminorm[20] nor a paranorm.
  • If a TVS has a bounded neighborhood of 0 then it is pseudometrizable; the converse is in general false.[14]
  • If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.[14]
  • If a TVS X has a convex bounded neighborhood of the origin then it is seminormable ; if in addition X is Hausdorff then it is normable.[14]

If X is a Hausdorff locally convex TVS then the following are equivalent:

  1. X is normable.
  2. X has a bounded neighborhood of the origin.
  3. the strong dual of X is normable.[21]
  4. the strong dual of X is metrizable.[21]

If X is Hausdorff locally convex TVS then X with the strong topology, (X, b(X, X')), is metrizable if and only if there exists a countable set of bounded subsets of X such that every bounded subset of X is contained in some element of .[22]

Metrically bounded sets and bounded sets

Suppose that (X, d) is a pseudometric space and BX. We say that B is metrically bounded or d-bounded if there exists a real number R > 0 such that d(x, y) ≤ R for all x, yB; the smallest such R is then called the diameter or d-diameter of B.[14] If B is bounded in a pseudometrizable TVS X then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.[14]

Properties of pseudometrizable TVS

Theorem[23]  All infinite-dimensional separable complete metrizable TVS are homeomorphic.

Every metrizable locally convex TVS is a quasibarrelled space,[24] bornological space, and a Mackey space.

If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is infrabarreled, if and only if it is barreled.[25]

If X is a complete pseudo-metrizable TVS and M is a closed vector subspace of X, then X/M is complete.[10]

The strong dual of a locally convex metrizable TVS is a webbed space.[26]

If (X, 𝜏) and (X, 𝜐) are complete metrizable TVSs and if 𝜐 is coarser than 𝜏 then 𝜏 = 𝜐.[27] This is no longer true if either one of these metrizable TVSs is not complete.[28]

Completeness

Recall that every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If X is a metrizable TVS and d is a metric that defines X's topology, then its possible that X is complete as a TVS (i.e. relative to its uniformity) but the metric d is not a complete metric (such metrics exist even for X = ℝ). Thus, if X is a TVS whose topology is induced by a pseudometric d, then the notion of completeness of X (as a TVS) and the notion of completeness of the pseudometric space (X, d) are not always equivalent. The next theorem gives a condition for when they are equivalent:

Theorem[29]  If X is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric d, then d is a complete pseudometric on X if and only if X is complete as a TVS.

Theorem[30]  If X is a TVS whose topology is induced by a paranorm p, then X is complete if and only if for every sequence (xi)
i=1
in X, if
i=1
p(xi) < ∞
then
i=1
xi
converges in X.

Subsets and subsequences

  • In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[31]
  • If d is a translation invariant metric on a vector space X, then d(nx, 0) ≤ nd(x, 0) for all xX and every positive integer n.[32]
  • If (xi)
    i=1
    is a null sequence (i.e. it converges to the origin) in a metrizable TVS then there exists a sequence (ri)
    i=1
    of positive real numbers diverging to such that (rixi)
    i=1
    → 0
    .[32]

Banach-Saks theorem[33]  If (xn)
n=1
is a sequence in a locally convex metrizable TVS (X, 𝜏) that converges weakly to some xX, then there exists a sequence y = (yi)
i=1
in X such that yx in (X, 𝜏) and each yi is a convex combination of finitely many xn.

Mackey's countability condition[14]  Suppose that X is a locally convex metrizable TVS and that (Bi)
i=1
is a countable sequence of bounded subsets of X. Then there exists a bounded subset B of X and a sequence (ri)
i=1
of positive real numbers such that Biri B for all i.

Linear maps

  • If X is a pseudometrizable TVS and A maps bounded subsets of X to bounded subsets of Y, then A is continuous.[14]
  • Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.[34]
    • Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[34]

If F : XY is a linear map between TVSs and X is metrizable then the following are equivalent:

  1. F is continuous;
  2. F is a (locally) bounded map (i.e. F maps (von Neumann) bounded subsets of X to bounded subsets of Y);[11]
  3. F is sequentially continuous;[11]
  4. the image under F of every null sequence in X is a bounded set;[11]
    • Recall that a null sequence is a sequence that converges to the origin.
  5. F maps null sequences to null sequences;

Hahn-Banach extension property

Let X be a TVS. Say that a vector subspace M of X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X.[22] Say that X has the Hahn-Banach extension property (HBEP) if every vector subspace of X has the extension property.[22]

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

Theorem[22] (Kalton)  Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.

If a vector space X has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.[22]

gollark: If you don't have a consistent format for dates and whatnot, bad things happen.
gollark: If everything was UTF-8 strings and then application-parsed they would all be handled slightly differently by each library and it would be bad.
gollark: Also, probably more interoperability.
gollark: See above.
gollark: Hacks like having your lists be strings with items separated by commas are very potatos.

See also

References

  1. Narici 2011, pp. 1-18.
  2. Narici 2011, pp. 37-40.
  3. Swartz 1992, p. 15.
  4. Wilansky 2013, p. 17.
  5. Wilansky 2013, p. 15.
  6. Schechter pp.689-691
  7. Wilansky 2013, pp. 15-18.
  8. Schechter 1996, p. 692.
  9. Schechter 1996, p. 691.
  10. Narici 2011, pp. 91-95.
  11. Jarchow 1981, pp. 38-42.
  12. Schechter pp. 689-691
  13. Narici 2011, p. 123.
  14. Narici 2011, pp. 156-175.
  15. Schechter 1996, p. 487.
  16. Schechter 1996, pp. 692-693.
  17. Schechter p. 706
  18. Narici 2011, pp. 115-154.
  19. Wilansky 2013, pp. 15-16.
  20. Schaefer 1999, pp. 91-92.
  21. Treves 2006, p. 201.
  22. Narici 2011, pp. 225-273.
  23. Wilansky 2013, p. 57.
  24. Jarchow 1981, p. 222.
  25. Schaefer & Wolff 1999, p. 153.
  26. Narici 2011, pp. 459-483.
  27. Köthe 1969, p. 168.
  28. Wilansky 2013, p. 59.
  29. Narici 2011, pp. 47-50.
  30. Wilansky 2013, pp. 56-57.
  31. Narici 2011, pp. 172-173.
  32. Rudin 1991, p. 22.
  33. Rudin 1991, p. 67.
  34. Narici 2011, p. 125.
  • 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)
  • Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
  • 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)
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (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)
  • 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)
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • 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)
  • 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)
  • 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)
  • 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)
  • Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.