Questions: Orthonormal Bases in Hilbert Spaces

Question 1

A student claims: 'If {eᵢ} is an orthonormal basis for Hilbert space H, then every f ∈ H equals a finite sum Σᵢ₌₁ᴺ cᵢeᵢ for some N.' What is wrong with this?

• A. The coefficients cᵢ are not computable in general, so the sum cannot be constructed
• B. In infinite-dimensional Hilbert spaces, every f equals an INFINITE series Σᵢ cᵢeᵢ that converges in the H-norm — not necessarily a finite sum
• C. The claim is correct for separable Hilbert spaces but fails for non-separable ones
• D. f must be written as an integral, not a sum, because the basis may be uncountable

Answer: (B)

This is the key conceptual shift from finite to infinite dimensions. In ℝⁿ, every vector is an exact finite combination of basis vectors. In an infinite-dimensional Hilbert space, 'basis' means the span is DENSE — every vector can be approximated arbitrarily well by finite linear combinations, but the exact representation requires an infinite series converging in norm. The partial sums Σᵢ₌₁ᴺ ⟨f,eᵢ⟩eᵢ approach f in the norm as N → ∞, but no finite N suffices for most f.

Question 2

Why does Fourier analysis work — that is, why can every square-integrable function on [0,1] be represented by its Fourier series?

• A. Because the Weierstrass approximation theorem guarantees trigonometric polynomials approximate all continuous functions
• B. Because the Fourier transform is an invertible operation on L² functions
• C. Because every L² function can be recovered pointwise from its Fourier coefficients
• D. Because the trigonometric functions {1, cos(2πnx), sin(2πnx)} form an orthonormal basis for L²([0,1]) — the Fourier expansion IS the basis expansion, with convergence in L² norm

Answer: (D)

Fourier analysis is not analogous to Hilbert space theory — it IS Hilbert space theory applied to L²([0,1]). The inner product ⟨f,g⟩ = ∫₀¹ f(x)g(x)dx makes L² a Hilbert space. The trig functions are orthonormal under this inner product and their span is dense in L², making them an orthonormal basis. The Fourier coefficient ĉₙ = ∫f(x)eₙ(x)dx is exactly ⟨f,eₙ⟩, and f = Σ ⟨f,eₙ⟩eₙ is the basis expansion formula. Convergence is in L² norm, not pointwise.

Question 3

An orthonormal system {eᵢ} in a Hilbert space is an orthonormal basis if and mainly if its span equals H — most element of H can be written as a finite linear combination of the eᵢ.

• T. True
• F. False

Answer: False

An orthonormal basis requires that the span is DENSE in H — every element can be approximated arbitrarily well by finite linear combinations, but not necessarily written as a finite combination exactly. 'Span equals H' is the finite-dimensional notion of basis. In infinite-dimensional spaces, the correct condition is density of the span (equivalently, maximality: no unit vector exists that is orthogonal to every element of the set). The expansion f = Σ⟨f,eᵢ⟩eᵢ is an infinite series converging in norm, not a finite sum.

Question 4

In a separable Hilbert space, every orthonormal basis is countable.

• T. True
• F. False

Answer: True

Separability means H has a countable dense subset. If the orthonormal basis were uncountable, then for any two distinct basis elements eᵢ, eⱼ, we have ‖eᵢ − eⱼ‖² = ‖eᵢ‖² − 2⟨eᵢ,eⱼ⟩ + ‖eⱼ‖² = 2. So the open balls of radius 1/√2 around each basis element are pairwise disjoint, and each must contain a point from the countable dense subset — requiring uncountably many such points. This contradicts countability. Hence the basis must be countable.

Question 5

What is the relationship between Fourier series and orthonormal bases in Hilbert space theory — and why is this not merely an analogy?

Model answer: The Fourier basis {1, cos(2πnx), sin(2πnx)} is literally an orthonormal basis for L²([0,1]) — the Hilbert space of square-integrable functions with inner product ⟨f,g⟩ = ∫f(x)g(x)dx. The trig functions are orthonormal under this inner product, and their span is dense in L² (L² is separable and this system is maximal). The Fourier expansion f = Σcₙeₙ with cₙ = ⟨f,eₙ⟩ is exactly the Hilbert space basis expansion formula. Convergence is in the L² norm. This is not an analogy — Fourier analysis is Hilbert space theory instantiated in L².

Understanding this relationship clarifies both directions. Hilbert space theory explains WHY Fourier series converge (in L² norm) and represent L² functions exactly. And Fourier analysis is the motivating historical example that guided the development of the abstract theory. Parseval's identity ‖f‖² = Σ|cₙ|² is Bessel's equality for a complete orthonormal system.