Hilbert Spaces — Mathematics Knowledge System

Overview

A Hilbert space is a complete inner product space — a Banach space whose norm arises from an inner product. The inner product endows the space with notions of angle and orthogonality, enabling powerful geometric methods. Hilbert spaces are the natural setting for quantum mechanics, Fourier analysis, and the study of self-adjoint operators. The Riesz representation theorem and Parseval's identity are cornerstones of the theory.

Intuition

An inner product is a way to measure both length and angle. In R^n, the dot product x·y = sum x_i y_i encodes both: ||x|| = sqrt(x·x) and cos(theta) = x·y/(||x||||y||). A Hilbert space extends this to infinite-dimensional settings. Orthogonality — the analogue of perpendicularity — allows us to decompose vectors into independent components, just as we write any vector in R^3 as a combination of i, j, k. Parseval's identity says that this decomposition preserves the total 'energy' (squared norm).

Formal Definition

Definition

An inner product space is a vector space H over F (R or C) with an inner product <·,·>: H x H -> F satisfying conjugate symmetry, linearity in the first argument, and positive definiteness. The associated norm is ||x|| = sqrt(<x,x>). A Hilbert space is an inner product space that is complete with respect to this norm. Two vectors x,y are orthogonal (x perp y) if <x,y> = 0.

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

Conjugate symmetry

\langle \alpha x + \beta z, y \rangle = \alpha\langle x,y\rangle + \beta\langle z,y\rangle

Linearity in first argument

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

Induced norm

|\langle x, y \rangle| \le \|x\|\,\|y\|

Cauchy-Schwarz inequality

\sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2

Parseval's identity

Notation

Notation	Meaning
$\langle x, y \rangle$	Inner product of x and y
$x \perp y$	x is orthogonal to y: <x,y>=0
$M^\perp$	Orthogonal complement of M
$\hat{x}_n = \langle x, e_n \rangle$	Fourier coefficient of x with respect to e_n

Properties

Parallelogram law

\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2) for all x,y in H.

Polarisation identity (real case)

\langle x,y \rangle = \tfrac{1}{4}(\|x+y\|^2 - \|x-y\|^2)

Theorems

Theorem 3.1: Cauchy-Schwarz Inequality

For all x, y in an inner product space H: |<x,y>| <= ||x|| ||y||, with equality if and only if x and y are linearly dependent.

Theorem 3.2: Riesz Representation Theorem

Let H be a Hilbert space and f: H -> F a bounded linear functional. Then there exists a unique y in H such that f(x) = <x,y> for all x in H, and ||f|| = ||y||. Thus H* is isometrically isomorphic to H.

Theorem 3.3: Theorem 3.3

Let (e_n) be a complete orthonormal system (orthonormal basis) in a separable Hilbert space H. Then for every x in H: x = sum_{ n=1 }^inf <x,e_n> e_n and ||x||^2 = sum_{ n=1 }^inf |<x,e_n>|^2.

Theorem 3.4: Theorem 3.4

Let M be a closed subspace of a Hilbert space H. Then H = M + M^perp (orthogonal direct sum): every x in H has a unique decomposition x = P_M x + P_{ M^perp } x where P_M is the orthogonal projection onto M.

Worked Examples

1
If y = 0, both sides are 0 and equality holds. Assume y != 0.
2
For any scalar t in F, consider the expansion of ||x - t y||^2 >= 0:
$0 \le \|x - ty\|^2 = \|x\|^2 - t\langle x,y\rangle - \bar{t}\langle y,x\rangle + |t|^2\|y\|^2$
3
Choose t = <x,y>/||y||^2 to minimise the right side. Substituting:
$0 \le \|x\|^2 - \frac{|\langle x,y\rangle|^2}{\|y\|^2}$
4
Rearranging gives |<x,y>|^2 <= ||x||^2 ||y||^2, hence |<x,y>| <= ||x|| ||y||.

✓ Answer

|<x,y>| <= ||x|| ||y||, with equality iff x = ty for some scalar t.

Practice Problems

Mediumproof writing

Prove that orthogonal complement M^perp of any subset M of a Hilbert space H is a closed subspace.

Mediumfill in blank

In a Hilbert space H with orthonormal basis (e_n), the representation x = sum_{n} <x,e_n> e_n is called the ___ expansion, and the identity ||x||^2 = sum_n |<x,e_n>|^2 is called ___ identity.

Mediumfree response

Use the Riesz representation theorem to show that every Hilbert space H is isometrically isomorphic to its own dual H*.

Common Mistakes

Common Mistake

Thinking every Banach space is a Hilbert space.

A Banach space is a Hilbert space only if its norm satisfies the parallelogram law ||x+y||^2 + ||x-y||^2 = 2(||x||^2+||y||^2). For example, L^1[0,1] is Banach but not Hilbert.

Common Mistake

Confusing orthonormal set with orthonormal basis.

An orthonormal set has mutually orthogonal unit vectors. An orthonormal basis is additionally complete (maximal). Only a complete orthonormal system gives Parseval's identity.

Quiz

Historical Background

David Hilbert introduced his famous space l^2 of square-summable sequences in 1906 while studying integral equations. The abstract axiomatic definition of a Hilbert space was formulated by John von Neumann in 1928. Hilbert's student Erhard Schmidt had already developed orthogonalisation procedures (now known as Gram-Schmidt) and the geometric viewpoint. The Riesz representation theorem was proved by Frédéric Riesz in 1907 for L^2[0,1].

1906
Hilbert studies l^2 in the context of integral equations
David Hilbert
1907
Riesz and Fischer prove the completeness of L^2
Frédéric Riesz, Ernst Fischer
1928
Von Neumann gives the modern axiomatic definition of Hilbert spaces
John von Neumann
1932
Von Neumann's Mathematische Grundlagen der Quantenmechanik applies Hilbert spaces to quantum theory
John von Neumann

Summary

A Hilbert space is a complete inner product space: it combines geometric (angle, orthogonality) and analytic (completeness) structure.
The Cauchy-Schwarz inequality |<x,y>| <= ||x|| ||y|| is fundamental and implies continuity of the inner product.
The Riesz representation theorem identifies H with its dual H*: every bounded linear functional is given by an inner product.
An orthonormal basis (complete orthonormal system) gives Parseval's identity: ||x||^2 = sum_n |<x,e_n>|^2.
The projection theorem: every closed subspace M has an orthogonal complement M^perp, and H = M direct sum M^perp.

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 4
BookConway, J. — A Course in Functional Analysis, Chapter 1
Websiteen.wikipedia.org