Hilbert space

Hilbert space is a generalization of Euclidean space that admits infinite dimension . Named after David Hilbert .

The most important object of research in Hilbert space is linear operators . The very concept of Hilbert space was formed in the works of Hilbert and Schmidt on the theory of integral equations , and an abstract definition was given in the works of von Neumann , Ries and Stone on the theory of Hermitian operators .

Content

Definition

Hilbert space is a linear (vector) space (over the field of real or complex numbers) in which ^[1] :

a rule is indicated that allows you to define for any two elements of space $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ their scalar product $(x,y)$ ${\ displaystyle (x, y)}$ ;
This rule satisfies the following requirements:
- $(y,x)=(x,y)$ ${\ displaystyle (y, x) = (x, y)}$ (translational law in real Hilbert space) or $(y,x)={\overline {(x,y)}}$ ${\ displaystyle (y, x) = {\ overline {(x, y)}}}$ (moving law in a complex Hilbert space, a bar means a sign of complex conjugation) ^[2] ;
- $(x,y+z)=(x,y)+(x,z)$ ${\ displaystyle (x, y + z) = (x, y) + (x, z)}$ (distribution law);
- $(\lambda x,y)=\lambda (x,y)$ ${\ displaystyle (\ lambda x, y) = \ lambda (x, y)}$ for any real number $\lambda$ ${\ displaystyle \ lambda}$ ;
- $(x,x)>0$ ${\ displaystyle (x, x)> 0}$ at $x\neq 0$ ${\ displaystyle x \ neq 0}$ and $(x,x)=0$ ${\ displaystyle (x, x) = 0}$ at $x=0$ ${\ displaystyle x = 0}$ .
which is complete with respect to this scalar product of the metric $d(x,y)=\|x-y\|={\sqrt {(x-y,x-y)}}$ ${\ displaystyle d (x, y) = \ | xy \ | = {\ sqrt {(xy, xy)}}}$ . If the condition for the completeness of space is not fulfilled, then we speak of a pre-Hilbert space . However, most of the known (used) spaces are either complete or can be replenished.

Thus, a Hilbert space is a Banach space (a complete normed space) whose norm is generated by a positive definite scalar product and is defined as $\|x\|={\sqrt {(x,x)}}$ ${\ displaystyle \ | x \ | = {\ sqrt {(x, x)}}}$

A norm in an arbitrary normed space can be generated by some scalar product if and only if the following parallelogram equality (identity) holds :

(\forall x,y\in H)\quad \|x+y\|^{2}+\|x-y\|^{2}=2(\|x\|^{2}+\|y\|^{2}).

{\ displaystyle (\ forall x, y \ in H) \ quad \ | x + y \ | ^ {2} + \ | xy \ | ^ {2} = 2 (\ | x \ | ^ {2} + \ | y \ | ^ {2}).}

If a Banach space satisfying the parallelogram identity is real, then the scalar product corresponding to its norm is given by

(x,y)=\left\|{\dfrac {x+y}{2}}\right\|^{2}-\left\|{\dfrac {x-y}{2}}\right\|^{2}.

{\ displaystyle (x, y) = \ left \ | {\ dfrac {x + y} {2}} \ right \ | ^ {2} - \ left \ | {\ dfrac {xy} {2}} \ right \ | ^ {2}.}

If this space is complex, then the scalar product corresponding to its norm is given by the equality

(x,y)=\left\|{\dfrac {x+y}{2}}\right\|^{2}-\left\|{\dfrac {x-y}{2}}\right\|^{2}+i\left\|{\dfrac {x+iy}{2}}\right\|^{2}-i\left\|{\dfrac {x-iy}{2}}\right\|^{2}

{\ displaystyle (x, y) = \ left \ | {\ dfrac {x + y} {2}} \ right \ | ^ {2} - \ left \ | {\ dfrac {xy} {2}} \ right \ | ^ {2} + i \ left \ | {\ dfrac {x + iy} {2}} \ right \ | ^ {2} -i \ left \ | {\ dfrac {x-iy} {2}} \ right \ | ^ {2}}

(polarization identity).

The Cauchy-Bunyakovsky inequality. Orthogonality

In the Hilbert space, the Cauchy-Bunyakovsky inequality is important:

|(x,y)|\leqslant \|x\|\|y\|

{\ displaystyle | (x, y) | \ leqslant \ | x \ | \ | y \ |}

.

This inequality in the case of a real Hilbert space makes it possible to determine the angle $\varphi$ ${\ displaystyle \ varphi}$ between two elements x and y according to the following formula

\cos \varphi ={\frac {(x,y)}{\|x\|\|y\|}}

{\ displaystyle \ cos \ varphi = {\ frac {(x, y)} {\ | x \ | \ | y \ |}}}

.

In particular, if the scalar product is zero $(x,y)=0$ ${\ displaystyle (x, y) = 0}$ , and the elements themselves are nonzero, then the angle between these elements is equal to $90^{\circ }$ ${\ displaystyle 90 ^ {\ circ}}$ , which corresponds to the orthogonality of the elements x and y. Note that the concept of orthogonality is also introduced in a complex Hilbert space using the relation $(x,y)=0$ ${\ displaystyle (x, y) = 0}$ . To indicate the orthogonality of the elements, use the symbol $\perp$ ${\ displaystyle \ perp}$ . Two subsets $M$ ${\ displaystyle M}$ and $N$ ${\ displaystyle N}$ Hilbert spaces are orthogonal $(M\perp N)$ ${\ displaystyle (M \ perp N)}$ if any two elements $f\in M$ ${\ displaystyle f \ in M}$ , $g\in N$ ${\ displaystyle g \ in N}$ orthogonal.

For pairwise orthogonal vectors, the Pythagorean theorem (generalized) holds:

\left\|\sum _{i}x_{i}\right\|^{2}=\sum _{i}\|x_{i}\|^{2}

{\ displaystyle \ left \ | \ sum _ {i} x_ {i} \ right \ | ^ {2} = \ sum _ {i} \ | x_ {i} \ | ^ {2}}

.

The set of all elements of space orthogonal to some subset $A$ ${\ displaystyle A}$ is a closed linear manifold (subspace) and is called the orthogonal complement of this set.

A subset of elements is called an orthonormal system if any two elements of the set are orthogonal and the norm of each element is unity.

Bases and dimension of a Hilbert space

The system of vectors of a Hilbert space is complete if it generates the whole space, that is, if an arbitrary element of space can be arbitrarily exactly approximated in norm by linear combinations of elements of this system. If a countable complete system of elements exists in space, then the space is separable - that is, there is a countable everywhere dense set whose closure in the space metric coincides with the whole space.

This complete system $\{e_{i}\}$ ${\ displaystyle \ {e_ {i} \}}$ is a basis if each element of space can be represented as a linear combination of elements of this system and, moreover, uniquely. It should be noted that in the general case of Banach spaces it does not follow from the completeness and linear independence of the elements of the system that this is a basis. However, in the case of separable Hilbert spaces, the complete orthonormal system $\{e_{i}\}$ ${\ displaystyle \ {e_ {i} \}}$ is the basis. In order for an orthonormal system to be complete in a separable Hilbert space, it is necessary and sufficient that there is no nonzero element orthogonal to all elements of the orthonormal system. So for every item $f$ ${\ displaystyle f}$ space expansion takes place in an orthonormal basis $\{e_{i}\}$ ${\ displaystyle \ {e_ {i} \}}$ :

$f=\sum _{i=1}^{\infty }\alpha _{i}e_{i}=\sum _{i=1}^{\infty }(f,e_{i})e_{i}$ ${\ displaystyle f = \ sum _ {i = 1} ^ {\ infty} \ alpha _ {i} e_ {i} = \ sum _ {i = 1} ^ {\ infty} (f, e_ {i}) e_ {i}}$

Decomposition coefficients $\alpha _{i}=(f,e_{i})$ ${\ displaystyle \ alpha _ {i} = (f, e_ {i})}$ called the Fourier coefficients. Moreover, for the norm of the element the Parseval equality is satisfied :

$\|f\|^{2}=\sum _{i=1}^{\infty }|(f,e_{i})|^{2}$ ${\ displaystyle \ | f \ | ^ {2} = \ sum _ {i = 1} ^ {\ infty} | (f, e_ {i}) | ^ {2}}$

All orthonormal bases in a Hilbert space have the same cardinality, which allows us to define the dimension of a Hilbert space as the dimension of an arbitrary orthonormal basis (orthogonal dimension). Hilbert space is separable if and only if it has a countable dimension.

The dimension of space can also be defined as the smallest of cardinalities of subsets of a Hilbert space $H$ ${\ displaystyle H}$ for which the closure of the linear shell coincides with $H$ ${\ displaystyle H}$ .

Any two Hilbert spaces having the same dimension are isomorphic . In particular, any two infinite-dimensional separable Hilbert spaces are isomorphic to each other and to the space $\ell ^{2}$ ${\ displaystyle \ ell ^ {2}}$ .

There are inseparable Hilbert spaces — spaces in which there is no countable basis ^[3] . In particular, an example of an inseparable space is interesting. $L^{2}$ ${\ displaystyle L ^ {2}}$ with special measure ^[4] .

Orthogonal expansions

Let be $L$ ${\ displaystyle L}$ - some subspace in the Hilbert space $H$ ${\ displaystyle H}$ . Then for any element $f\in H$ ${\ displaystyle f \ in H}$ the only decomposition is true $f=g+h$ ${\ displaystyle f = g + h}$ where $g\in L$ ${\ displaystyle g \ in L}$ , but $h\perp L$ ${\ displaystyle h \ perp L}$ . Element $g$ ${\ displaystyle g}$ called element projection $f$ ${\ displaystyle f}$ on $L$ ${\ displaystyle L}$ . Collection of elements $h$ ${\ displaystyle h}$ orthogonal to the subspace $L$ ${\ displaystyle L}$ forms a (closed) subspace $M$ ${\ displaystyle M}$ being the orthogonal complement of the subspace $L$ ${\ displaystyle L}$ .

They say that space $H$ ${\ displaystyle H}$ decomposed into the direct sum of subspaces $L$ ${\ displaystyle L}$ and $M$ ${\ displaystyle M}$ that is written as $H=L\oplus M$ ${\ displaystyle H = L \ oplus M}$ . Similarly, you can write $L=H\ominus M$ ${\ displaystyle L = H \ ominus M}$ .

The space of linear functionals

The space of linear continuous (bounded) functionals also forms a linear space and is called the conjugate space.

The following Riesz theorem on the general form of a bounded linear functional in a Hilbert space holds: for any bounded linear functional $f$ ${\ displaystyle f}$ on the hilbert space $H$ ${\ displaystyle H}$ there is a single vector $y\in H$ ${\ displaystyle y \ in H}$ such that $f(x)=(x,y)$ ${\ displaystyle f (x) = (x, y)}$ for anyone $x\in H$ ${\ displaystyle x \ in H}$ . Moreover, the norm of the linear functional $f$ ${\ displaystyle f}$ coincides with the norm of the vector $y$ ${\ displaystyle y}$ :

$\|f\|=\sup _{\|x\|=1}|f(x)|={\sqrt {(y,y)}}$ ${\ displaystyle \ | f \ | = \ sup _ {\ | x \ | = 1} | f (x) | = {\ sqrt {(y, y)}}}$ .

It follows from the theorem that the space of linear bounded functionals over a Hilbert space $H$ ${\ displaystyle H}$ isomorphic to space itself $H$ ${\ displaystyle H}$ .

Linear operators in Hilbert spaces

Linear operator $A$ ${\ displaystyle A}$ can be represented in this basis by matrix elements in a unique way: $a_{ij}=(Ae_{i},e_{j})$ ${\ displaystyle a_ {ij} = (Ae_ {i}, e_ {j})}$ .

Linear operator $A^{*}$ ${\ displaystyle A ^ {*}}$ called conjugate to the operator $A$ ${\ displaystyle A}$ if for any elements $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ equality holds $(Ax,y)=(x,A^{*}y)$ ${\ displaystyle (Ax, y) = (x, A ^ {*} y)}$ . The norm of the adjoint operator is equal to the norm of the operator itself.

A bounded linear operator is called self-adjoint ( symmetric ) if $A^{*}=A$ ${\ displaystyle A ^ {*} = A}$ .

Operator $P$ ${\ displaystyle P}$ defined on the whole space, which associates with each element its projection onto a certain subspace, is called the projection operator, (projection operator). A projector is an operator such that $P^{2}=P$ ${\ displaystyle P ^ {2} = P}$ . If, in addition, the projector $P$ ${\ displaystyle P}$ is a self-adjoint operator, it is also an orthogonal projector. The product of two design operators is design if and only if they commute: $P_{1}P_{2}=P_{2}P_{1}$ ${\ displaystyle P_ {1} P_ {2} = P_ {2} P_ {1}}$ .

Properties

Riesz representation theorem : for any orthonormal system of vectors ${\lbrace \phi _{i}\rbrace }_{i=1}^{\infty }$ ${\ displaystyle {\ lbrace \ phi _ {i} \ rbrace} _ {i = 1} ^ {\ infty}}$ in hilbert space $H$ ${\ displaystyle H}$ and numerical sequence ${\lbrace C_{i}\rbrace }_{i=1}^{\infty }$ ${\ displaystyle {\ lbrace C_ {i} \ rbrace} _ {i = 1} ^ {\ infty}}$ such that $\sum _{i=1}^{\infty }C_{i}^{2}<\infty$ ${\ displaystyle \ sum _ {i = 1} ^ {\ infty} C_ {i} ^ {2} <\ infty}$ , at $H$ ${\ displaystyle H}$ there is such an element $u$ ${\ displaystyle u}$ , what $C_{i}=\left(u,\phi _{i}\right)$ ${\ displaystyle C_ {i} = \ left (u, \ phi _ {i} \ right)}$ and ${\left\Vert u\right\Vert }^{2}=\sum _{i=1}^{\infty }{\left(u,\phi _{i}\right)}^{2}$ ${\ displaystyle {\ left \ Vert u \ right \ Vert} ^ {2} = \ sum _ {i = 1} ^ {\ infty} {\ left (u, \ phi _ {i} \ right)} ^ { 2}}$ .
Hilbert spaces generate strictly normed spaces .

Examples

Euclidean space .
Space $\ell ^{2}$ ${\ displaystyle \ ell ^ {2}}$ . Its points are infinite sequences of real numbers $x=\{x_{n}\}_{n=1}^{\infty }$ ${\ displaystyle x = \ {x_ {n} \} _ {n = 1} ^ {\ infty}}$ for which the series converges $\sum _{n=1}^{\infty }|x_{n}|^{2}$ ${\ displaystyle \ sum _ {n = 1} ^ {\ infty} | x_ {n} | ^ {2}}$ . The scalar product on this space is given by the equality
$(x,y)=\sum _{n=1}^{\infty }x_{n}y_{n}$ ${\ displaystyle (x, y) = \ sum _ {n = 1} ^ {\ infty} x_ {n} y_ {n}}$ .
Space $L^{2}[a,b]$ ${\ displaystyle L ^ {2} [a, b]}$ measurable functions with real values on the interval $[a,b]$ ${\ displaystyle [a, b]}$ with Lebesgue integrable squares - that is, such that the integral
$\int \limits _{a}^{b}\!|f|^{2}\,dx$ ${\ displaystyle \ int \ limits _ {a} ^ {b} \! | f | ^ {2} \, dx}$

is defined and finite, moreover, functions that differ from each other on the set of measure zero are identified among themselves (that is, formally,

L^{2}[a,b]

{\ displaystyle L ^ {2} [a, b]}

there is a corresponding set of equivalence classes). The scalar product on this space is given by the equality

(f,g)=\int \limits _{a}^{b}\!f{g}\,dx

{\ displaystyle (f, g) = \ int \ limits _ {a} ^ {b} \! f {g} \, dx}

.

For spaces $\ell ^{2}$ ${\ displaystyle \ ell ^ {2}}$ and $L^{2}[a,b]$ ${\ displaystyle L ^ {2} [a, b]}$ over the field of complex numbers, sequences of complex numbers and complex-valued functions, the definition of a scalar product differs only in the complex conjugacy of the second factor:

(x,y)=\sum _{n=1}^{\infty }x_{n}{\overline {y}}_{n}

{\ displaystyle (x, y) = \ sum _ {n = 1} ^ {\ infty} x_ {n} {\ overline {y}} _ {n}}

;

(f,g)=\int \limits _{a}^{b}\!f{\overline {g}}\,dx

{\ displaystyle (f, g) = \ int \ limits _ {a} ^ {b} \! f {\ overline {g}} \, dx}

.

Notes

↑ Shilov G.E. Mathematical analysis. Special course. - M .: Fizmatlit, 1961. - C. 181
↑ Shilov G.E. Mathematical analysis. Special course. - M .: Fizmatlit, 1961. - C. 253
↑ Konstantinov R.V. Lectures on functional analysis. - M .: MIPT, 2009. - C. 129
↑ Reed, M., Simon, B. Methods of modern mathematical physics. Volume 1. Functional analysis. - M.: Mir, 1977 .-- C. 82

Literature

Halmosh P. , Hilbert space in problems , Translation from English by I. D. Novikov and T. V. Sokolovskaya; under the editorship of R. A. Minlosa. - M .: Mir Publishing House, 1970. - 352 p.
Moren K. , Methods of Hilbert space. - M.: Mir, 1965 .-- 570 p.

[1] Shilov G.E. Mathematical analysis. Special course. - M .: Fizmatlit, 1961. - C. 181

[2] Shilov G.E. Mathematical analysis. Special course. - M .: Fizmatlit, 1961. - C. 253

[3] Konstantinov R.V. Lectures on functional analysis. - M .: MIPT, 2009. - C. 129

[4] Reed, M., Simon, B. Methods of modern mathematical physics. Volume 1. Functional analysis. - M.: Mir, 1977 .-- C. 82