Mathematics.

Functional Analysis

Hilbert Spaces

Real Analysis90 minDifficulty8 out of 10

You should know: banach spaces, inner product spaces, fourier series

Overview

A Hilbert space is a complete inner product space — an infinite-dimensional generalisation of Euclidean space. The inner product provides notions of angle, orthogonality, and projection that go beyond a bare norm. The most important examples are L^2(\mu) (square-integrable functions) and \ell^2 (square-summable sequences). Hilbert spaces are the natural setting for quantum mechanics, Fourier analysis, and the theory of PDEs.

Intuition

Think of a Hilbert space as an infinite-dimensional Euclidean space where you can still talk about perpendicular directions and projections. Just as in \mathbb{R}^n you can decompose a vector into components along an orthonormal basis, in a Hilbert space you can expand a function in an orthonormal series (like a Fourier series) and the Pythagorean theorem still holds. The completeness ensures these infinite series converge.

Formal Definition

Definition

A Hilbert space is a vector space H over \mathbb{R} or \mathbb{C} equipped with an inner product \langle \cdot, \cdot \rangle that is complete in the induced norm \|x\| = \sqrt{\langle x, x \rangle}.

⟨x,y⟩:H×H→F,∥x∥=⟨x,x⟩.\langle x, y \rangle: H \times H \to \mathbb{F}, \quad \|x\| = \sqrt{\langle x, x \rangle}.
Inner product and induced norm
⟨x,y⟩=⟨y,x⟩‾,⟨αx+βy,z⟩=α⟨x,z⟩+β⟨y,z⟩,⟨x,x⟩≥0.\langle x, y \rangle = \overline{\langle y, x \rangle}, \quad \langle \alpha x + \beta y, z \rangle = \alpha \langle x, z \rangle + \beta \langle y, z \rangle, \quad \langle x, x \rangle \geq 0.
Inner product axioms
L2(μ):⟨f,g⟩=∫fg‾ dμ,ℓ2:⟨(xn),(yn)⟩=∑n=1∞xnyn‾.L^2(\mu): \langle f, g \rangle = \int f \overline{g}\, d\mu, \qquad \ell^2: \langle (x_n), (y_n) \rangle = \sum_{n=1}^{\infty} x_n \overline{y_n}.
Key examples
∥x+y∥2=∥x∥2+∥y∥2(x⊥y, i.e. ⟨x,y⟩=0).\|x + y\|^2 = \|x\|^2 + \|y\|^2 \quad (x \perp y, \text{ i.e. } \langle x,y\rangle = 0).
Pythagorean theorem

Properties

Cauchy-Schwarz inequality

∣⟨x,y⟩∣≤∥x∥∥y∥, with equality iff x and y are proportional.|\langle x, y \rangle| \leq \|x\| \|y\|, \text{ with equality iff } x \text{ and } y \text{ are proportional.}

Parallelogram law

∥x+y∥2+∥x−y∥2=2(∥x∥2+∥y∥2).\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2).

Projection theorem

ForanyclosedsubspaceM⊆H,everyx∈Hhasauniqueorthogonaldecompositionx=PMx+(x−PMx),wherePMx∈Mand(x−PMx)⊥M.For any closed subspace M \subseteq H, every x \in H has a unique orthogonal decomposition x = P_M x + (x - P_M x), where P_M x \in M and (x - P_M x) \perp M.

Riesz Representation Theorem

Everyboundedlinearfunctionalϕ:H→Fhastheformϕ(x)=⟨x,y⟩forauniquey∈H.ThusH≅H∗.Every bounded linear functional \phi: H \to \mathbb{F} has the form \phi(x) = \langle x, y \rangle for a unique y \in H. Thus H \cong H^*.

Worked Examples

  1. Compute the inner product \langle e_n, e_m \rangle = \frac{1}{2\pi} \int_0^{2\pi} e^{inx} e^{-imx}\, dx = \frac{1}{2\pi} \int_0^{2\pi} e^{i(n-m)x}\, dx.

    ⟨en,em⟩=12π∫02πei(n−m)x dx.\langle e_n, e_m \rangle = \frac{1}{2\pi} \int_0^{2\pi} e^{i(n-m)x}\, dx.

  2. If n \neq m, this integral equals \frac{1}{2\pi} \cdot \frac{e^{i(n-m)2\pi}-1}{i(n-m)} = 0.

    n≠m:⟨en,em⟩=0.n \neq m: \langle e_n, e_m \rangle = 0.

  3. If n = m, the integral equals \frac{1}{2\pi} \int_0^{2\pi} 1\, dx = 1.

    n=m:⟨en,en⟩=1.n = m: \langle e_n, e_n \rangle = 1.

  4. So \langle e_n, e_m \rangle = \delta_{nm} — the system is orthonormal.

Answer: The complex exponentials \frac{1}{\sqrt{2\pi}} e^{inx} are orthonormal in L^2([0,2\pi]).

Practice Problems

Difficulty 6/10

Prove the Cauchy-Schwarz inequality |\langle x, y \rangle| \leq \|x\| \|y\| in an inner product space.

Difficulty 7/10

State the Projection Theorem and explain how it generalises the notion of 'closest point' in \mathbb{R}^n.

Difficulty 8/10

Prove that every separable Hilbert space has a countable orthonormal basis (Hilbert basis).

Common Mistakes

Common Mistake

Every Banach space is a Hilbert space.

Not every Banach space has an inner product. L^1([0,1]) and C([0,1]) with sup-norm are Banach but not Hilbert (their norms fail the parallelogram law).

Common Mistake

An orthonormal set spans the Hilbert space.

An orthonormal set spans the space only if it is a Hilbert basis (the closed linear span is all of H). An orthonormal set may be incomplete.

Quiz

What distinguishes a Hilbert space from a general Banach space?
Which identity characterises norms coming from an inner product?

Summary

  • A Hilbert space is a complete inner product space — the infinite-dimensional analogue of Euclidean space.
  • Key examples: L^2(\mu) and \ell^2; every separable Hilbert space is isometrically isomorphic to \ell^2.
  • Orthogonal projections, the Projection Theorem, and the Riesz Representation Theorem are the central structural results.
  • An orthonormal basis (Hilbert basis) allows every element to be expanded as an infinite series with Parseval's identity holding.
  • Hilbert spaces are the foundation for quantum mechanics, Fourier analysis, and PDE theory.

References

  1. BookRudin, W. — Real and Complex Analysis (3rd ed.), McGraw-Hill, 1987.
  2. WebsiteWikipedia — Hilbert space