Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.[1]

For the scalar product or dot product of coordinate vectors, see dot product.

Geometric interpretation of the angle between two vectors defined using an inner product

Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.

An inner product naturally induces an associated norm, (|x| and |y| are the norms of x and y, in the picture) thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An (incomplete) space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.

Definition

In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C.

Formally, an inner product space is a vector space V over the field F together with an inner product, i.e., with a map

${\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}$

that satisfies the following three properties for all vectors x, y, z ∈ V and all scalars a ∈ F:[2][3]

• Conjugate symmetry:[Note 1]

  ${\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}}$

• Linearity in the first argument:[Note 2]

  ${\displaystyle {\begin{aligned}\langle ax,y\rangle &=a\langle x,y\rangle \\\langle x+y,z\rangle &=\langle x,z\rangle +\langle y,z\rangle \end{aligned}}}$

• Positive-definite:

  ${\displaystyle \langle x,x\rangle >0,\quad x\in V\setminus \{\mathbf {0} \}.}$

If the positive-definite condition is replaced by merely requiring that ${\displaystyle \langle x,x\rangle \geq 0}$ for all x, then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form ${\displaystyle \langle \cdot ,\cdot \rangle }$ is an inner product if and only if for all x, if ${\displaystyle \langle x,x\rangle =0}$ then x = 0.[4]

Elementary properties

Positive-definiteness and linearity, respectively, ensure that:

${\displaystyle {\begin{aligned}\langle x,x\rangle &=0\Rightarrow x=\mathbf {0} \\\langle \mathbf {0} ,\mathbf {0} \rangle &=\langle 0x,0x\rangle =0\langle x,0x\rangle =0\end{aligned}}}$

Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, since we have:

${\displaystyle \langle x,x\rangle ={\overline {\langle x,x\rangle }}\,.}$

Conjugate symmetry and linearity in the first variable imply

${\displaystyle {\begin{aligned}\langle x,ay\rangle &={\overline {\langle ay,x\rangle }}={\overline {a}}{\overline {\langle y,x\rangle }}={\overline {a}}\langle x,y\rangle \\\langle x,y+z\rangle &={\overline {\langle y+z,x\rangle }}={\overline {\langle y,x\rangle }}+{\overline {\langle z,x\rangle }}=\langle x,y\rangle +\langle x,z\rangle \,\end{aligned}}}$

that is, conjugate linearity in the second argument. So, an inner product is a sesquilinear form. Conjugate symmetry is also called Hermitian symmetry, and a conjugate-symmetric sesquilinear form is called a Hermitian form. While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a positive-definite Hermitian form.

This important generalization of the familiar square expansion follows:

${\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +\langle x,y\rangle +\langle y,x\rangle +\langle y,y\rangle \,.}$

These properties, constituents of the above linearity in the first and second argument:

${\displaystyle {\begin{aligned}\langle x+y,z\rangle &=\langle x,z\rangle +\langle y,z\rangle \,,\\\langle x,y+z\rangle &=\langle x,y\rangle +\langle x,z\rangle \end{aligned}}}$

are otherwise known as additivity.

In the case of F = R, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. So, an inner product on a real vector space is a positive-definite symmetric bilinear form. That is,

${\displaystyle {\begin{aligned}\langle x,y\rangle &=\langle y,x\rangle \\\Rightarrow \langle -x,x\rangle &=\langle x,-x\rangle \,,\end{aligned}}}$

and the binomial expansion becomes:

${\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle \,.}$

Alternative definitions, notations and remarks

A common special case of the inner product, the scalar product or dot product, is written with a centered dot ${\displaystyle a\cdot b}$.

Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨x, y⟩ as ⟨y | x⟩ (the bra–ket notation of quantum mechanics), respectively y^†x (dot product as a case of the convention of forming the matrix product AB as the dot products of rows of A with columns of B). Here the kets and columns are identified with the vectors of V and the bras and rows with the linear functionals (covectors) of the dual space V^∗, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature,[5] taking ⟨x, y⟩ to be conjugate linear in x rather than y. A few instead find a middle ground by recognizing both ⟨·, ·⟩ and ⟨· | ·⟩ as distinct notations differing only in which argument is conjugate linear.

There are various technical reasons why it is necessary to restrict the base field to R and C in the definition. Briefly, the base field has to contain an ordered subfield in order for non-negativity to make sense,[6] and therefore has to have characteristic equal to 0 (since any ordered field has to have such characteristic). This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, e.g., the algebraic numbers or the constructible numbers. However, in these cases when it is a proper subfield (i.e., neither R nor C) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces.

In some cases we need to consider non-negative semi-definite sesquilinear forms. This means that ⟨x, x⟩ is only required to be non-negative. We show how to treat these below.

Some examples

Real numbers

A simple example is the real numbers with the standard multiplication as the inner product

${\displaystyle \langle x,y\rangle$ :=xy.}

Euclidean vector space

More generally, the real n-space Rⁿ with the dot product is an inner product space, an example of a Euclidean vector space.

${\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle$ :=x^{\textsf {T}}y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},}

where x^T is the transpose of x.

Complex coordinate space

The general form of an inner product on Cⁿ is known as the Hermitian form and is given by

${\displaystyle \langle x,y\rangle$ :=y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},}

where M is any Hermitian positive-definite matrix and y^† is the conjugate transpose of y. For the real case this corresponds to the dot product of the results of directionally different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.

Hilbert space

The article on Hilbert spaces has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. An example of an inner product which induces an incomplete metric occurs with the space C([a, b]) of continuous complex valued functions f and g on the interval [a, b]. The inner product is

${\displaystyle \langle f,g\rangle$ :=\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.}

This space is not complete; consider for example, for the interval [−1, 1] the sequence of continuous "step" functions, { f_k}_k, defined by:

${\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}}$

This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function.

Random variables

For real random variables X and Y, the expected value of their product

${\displaystyle \langle X,Y\rangle$ :=\operatorname {E} (XY)}

is an inner product.[7][8][9] In this case, ⟨X, X⟩ = 0 if and only if Pr(X = 0) = 1 (i.e., X = 0 almost surely). This definition of expectation as inner product can be extended to random vectors as well.

Real matrices

For real square matrices of the same size, ⟨A, B⟩ := tr(AB^T) with transpose as conjugation

${\displaystyle \left(\langle A,B\rangle =\left\langle B^{\textsf {T}},A^{\textsf {T}}\right\rangle \right)}$

is an inner product.

Vector spaces with forms

On an inner product space, or more generally a vector space with a nondegenerate form (so an isomorphism V → V^∗) vectors can be sent to covectors (in coordinates, via transpose), so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.

Norm

Inner product spaces are normed vector spaces for the norm defined by

${\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.}$

As for every normed vector space, a inner product space is a metric space, for the distance defined by

${\displaystyle d(x,y)=\|y-x\|}$

Directly from the axioms of the inner product, one can prove that the axioms of a norm are satisfied, as well as the following properties.

Homogeneity

For x an element of V and r a scalar

${\displaystyle \|rx\|=|r|\,\|x\|.}$

Triangle inequality

For x, y elements of V

${\displaystyle \|x+y\|\leq \|x\|+\|y\|.}$

These two properties show that one has indeed a norm.

Cauchy–Schwarz inequality

For x, y elements of V

${\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}$

with equality if and only if x and y are linearly dependent. In the Russian mathematical literature, this inequality is also known as the Cauchy–Bunyakovsky–Schwarz inequality.

Polarization identity

The inner product can be retrieved from the norm by the polarization identity

${\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\operatorname {Re} \langle x,y\rangle ,}$

which is a form of the law of cosines.

Orthogonality

Two vectors are orthogonal if their inner product is zero.
In the case of Euclidean vector spaces, which are inner product spaces of finite dimension over the reals, the inner product allows defining the (non oriented) angle of two nonzero vectors by

${\displaystyle \angle (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}},}$

and

${\displaystyle 0\leq \angle (x,y)\leq \pi .}$

Pythagorean theorem

Whenever x, y are in V and ⟨x, y⟩ = 0, then

${\displaystyle \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.}$

The proof of the identity requires only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component.

The name Pythagorean theorem arises from the geometric interpretation in Euclidean geometry.

Parseval's identity

An induction on the Pythagorean theorem yields: if x₁, …, x_n are orthogonal vectors, that is, ⟨x_j, x_k⟩ = 0 for distinct indices j, k, then

${\displaystyle \sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.}$

Parallelogram law

For x, y elements of V,

${\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.}$

The parallelogram law is, in fact, a necessary and sufficient condition for the existence of a inner product corresponding to a given norm.

Ptolemy's inequality

For x, y, z elements of V,

${\displaystyle \|x-y\|\|z\|+\|y-z\|\|x\|\geq \|x-z\|\|y\|.}$

Ptolemy's inequality is, in fact, a necessary and sufficient condition for the existence of a inner product corresponding to a given norm. In detail, Isaac Jacob Schoenberg proved in 1952 that, given any real, seminormed space, if its seminorm is ptolemaic, then the seminorm is the norm associated to an inner product.[10]

Orthonormal sequences

Let V be a finite dimensional inner product space of dimension n. Recall that every basis of V consists of exactly n linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis {e₁, ..., e_n} is orthonormal if ⟨e_i, e_j⟩ = 0 for every i ≠ j and ⟨e_i, e_i⟩ = ||e_i|| = 1 for each i.

This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let V be any inner product space. Then a collection

${\displaystyle E=\left\{e_{\alpha }\right\}_{\alpha \in A}}$

is a basis for V if the subspace of V generated by finite linear combinations of elements of E is dense in V (in the norm induced by the inner product). We say that E is an orthonormal basis for V if it is a basis and

${\displaystyle \left\langle e_{\alpha },e_{\beta }\right\rangle =0}$

if α ≠ β and ⟨e_α, e_α⟩ = ||e_α|| = 1 for all α, β ∈ A.

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

Theorem. Any separable inner product space V has an orthonormal basis.

Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Any complete inner product space V has an orthonormal basis.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).

Proof

Recall that the dimension of an inner product space is the cardinality of a maximal orthonormal system that it contains (by Zorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system, but as we shall see, the converse need not hold. Observe that if G is a dense subspace of an inner product space H, then any orthonormal basis for G is automatically an orthonormal basis for H. Thus, it suffices to construct an inner product space H with a dense subspace G whose dimension is strictly smaller than that of H. Let K be a Hilbert space of dimension ℵ0 (for instance, K = l^2(N)). Let E be an orthonormal basis of K, so |E| = ℵ0. Extend E to a Hamel basis E ∪ F for K, where E ∩ F = ∅. Since it is known that the Hamel dimension of K is c, the cardinality of the continuum, it must be that |F| = c. Let L be a Hilbert space of dimension c (for instance, L = l^2(R)). Let B be an orthonormal basis for L, and let φ : F → B be a bijection. Then there is a linear transformation T : K → L such that Tf = φ( f ) for f ∈ F, and Te = 0 for e ∈ E. Let H = K ⊕ L and let G = {(k, Tk) : k ∈ K)} be the graph of T. Let Ḡ be the closure of G in H; we will show Ḡ = H. Since for any e ∈ E we have (e, 0) ∈ G, it follows that K ⊕ 0 ⊂ Ḡ. Next, if b ∈ B, then b = Tf for some f ∈ F ⊂ K, so ( f, b) ∈ G ⊂ Ḡ; since ( f, 0) ∈ Ḡ as well, we also have (0, b) ∈ Ḡ. It follows that 0 ⊕ L ⊂ Ḡ, so Ḡ = H, and G is dense in H. Finally, {(e, 0) : e ∈ E} is a maximal orthonormal set in G; if ${\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle }$ for all e ∈ E then certainly k = 0, so (k, Tk) = (0, 0) is the zero vector in G. Hence the dimension of G is |E| = ℵ0, whereas it is clear that the dimension of H is c. This completes the proof.

Parseval's identity leads immediately to the following theorem:

Theorem. Let V be a separable inner product space and {e_k}_k an orthonormal basis of V. Then the map

${\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}$

is an isometric linear map V → l² with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided l² is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let V be the inner product space C[−π, π]. Then the sequence (indexed on set of all integers) of continuous functions

${\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}$

is an orthonormal basis of the space C[−π, π] with the L² inner product. The mapping

${\displaystyle f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}$

is an isometric linear map with dense image.

Orthogonality of the sequence {e_k}_k follows immediately from the fact that if k ≠ j, then

${\displaystyle \int _{-\pi }^{\pi }e^{-i(j-k)t}\,\mathrm {d} t=0.}$

Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on [−π, π] with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces

Several types of linear maps A from an inner product space V to an inner product space W are of relevance:

• Continuous linear maps, i.e., A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals {||Ax||}, where x ranges over the closed unit ball of V, is bounded.
• Symmetric linear operators, i.e., A is linear and ⟨Ax, y⟩ = ⟨x, Ay⟩ for all x, y in V.
• Isometries, i.e., A is linear and ⟨Ax, Ay⟩ = ⟨x, y⟩ for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective. Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
• Isometrical isomorphisms, i.e., A is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.

Generalizations

Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.

Degenerate inner products

If V is a vector space and ⟨·, ·⟩ a semi-definite sesquilinear form, then the function:

${\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}$

makes sense and satisfies all the properties of norm except that ||x|| = 0 does not imply x = 0 (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient W = V/{x : ||x|| = 0}. The sesquilinear form ⟨·, ·⟩ factors through W.

This construction is used in numerous contexts. The Gelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.

Nondegenerate conjugate symmetric forms

Main article: Pseudo-Euclidean space

Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero x there exists some y such that ⟨x, y⟩ ≠ 0, though y need not equal x; in other words, the induced map to the dual space V → V^∗ is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism V → V^∗) and thus hold more generally.

Related products

The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a 1 × n covector with an n × 1 vector, yielding a 1 × 1 matrix (a scalar), while the outer product is the product of an m × 1 vector with a 1 × n covector, yielding an m × n matrix. Note that the outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear map W × V^∗ → Hom(V, W) sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map V^∗ × V → F given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.

The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.

As a further complication, in geometric algebra the inner product and the exterior (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the outer product (alternatively, wedge product). The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).

See also

• Space (mathematics)
• Semi-inner-product
• Normed vector space
• Energetic space
• Dual space
• Biorthogonal system
• Bilinear form

Notes

1. A bar over an expression denotes complex conjugation; e.g., ${\textstyle {\overline {x}}}$ is the complex conjugation of ${\textstyle x}$. For real values, ${\textstyle x={\overline {x}}}$ and conjugate symmetry is just symmetry.
2. By combining the linear in the first argument property with the conjugate symmetry property you get conjugate-linear in the second argument: ${\textstyle \langle x,by\rangle =\langle x,y\rangle {\overline {b}}}$. This is how the inner product was originally defined and is still used in some old-school math communities. However, all of engineering and computer science, and most of physics and modern mathematics now define the inner product to be linear in the second argument and conjugate-linear in the first argument because this is more compatible with several other conventions in mathematics. Notably, for any inner product, there is some hermitian, positive-definite matrix ${\textstyle M}$ such that ${\textstyle \langle x,y\rangle =x^{*}My}$. (Here, ${\textstyle x^{*}}$ is the conjugate transpose of ${\textstyle x}$.)

References

1. Moore, Gregory H. (1995). "The axiomatization of linear algebra: 1875-1940". Historia Mathematica. 22 (3): 262–303. doi:10.1006/hmat.1995.1025.
2. Jain, P. K.; Ahmad, Khalil (1995). "5.1 Definitions and basic properties of inner product spaces and Hilbert spaces". Functional Analysis (2nd ed.). New Age International. p. 203. ISBN 81-224-0801-X.
3. Prugovec̆ki, Eduard (1981). "Definition 2.1". Quantum Mechanics in Hilbert Space (2nd ed.). Academic Press. pp. 18ff. ISBN 0-12-566060-X.
4. Schaefer 1999, p. 44.
5. Emch, Gerard G. (1972). Algebraic Methods in Statistical Mechanics and Quantum Field Theory. New York: Wiley-Interscience. ISBN 978-0-471-23900-0.
6. Finkbeiner, Daniel T. (2013), Introduction to Matrices and Linear Transformations, Dover Books on Mathematics (3rd ed.), Courier Dover Publications, p. 242, ISBN 9780486279664.
7. Ouwehand, Peter (November 2010). "Spaces of Random Variables" (PDF). AIMS. Retrieved 2017-09-05.
8. Siegrist, Kyle (1997). "Vector Spaces of Random Variables". Random: Probability, Mathematical Statistics, Stochastic Processes. Retrieved 2017-09-05.
9. Bigoni, Daniele (2015). "Appendix B: Probability theory and functional spaces" (PDF). Uncertainty Quantification with Applications to Engineering Problems (PhD). Technical University of Denmark. Retrieved 2017-09-05.
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