Questions: Bessel's Inequality and Parseval's Identity

Question 1

For a Hilbert space vector x and an orthonormal system {eᵢ}, you compute Σ|⟨x, eᵢ⟩|² and find it is strictly less than ‖x‖². What can you conclude?

• A. The partial series has not converged yet; summing more terms will close the gap
• B. The orthonormal system does not form a complete basis for the Hilbert space
• C. The vector x was not correctly normalized before computing the coefficients
• D. The Fourier series for x diverges in this orthonormal system

Answer: (B)

A strict inequality Σ|⟨x, eᵢ⟩|² < ‖x‖² means the system is incomplete — x has a nonzero component in directions orthogonal to every eᵢ. The gap is not about truncation: even the full infinite sum fails to reach ‖x‖². Parseval's identity (equality) is equivalent to completeness of the orthonormal system.

Question 2

What is the fundamental reason Bessel's inequality (Σᵢ|⟨x, eᵢ⟩|² ≤ ‖x‖²) holds for any orthonormal system?

• A. Orthogonality forces the Fourier coefficients to sum to zero, bounding their squares
• B. The Cauchy-Schwarz inequality bounds each individual coefficient by ‖x‖, so their sum is bounded
• C. The squared norm ‖x − Sₙ‖² is always non-negative, which forces the sum of squared coefficients to be at most ‖x‖²
• D. Convergence of the Fourier series in norm implies the coefficients must be square-summable

Answer: (C)

Expanding ‖x − Sₙ‖² using orthonormality yields ‖x‖² − Σᵢ₌₁ⁿ|cᵢ|². Since any squared norm is ≥ 0, this gives Σᵢ₌₁ⁿ|cᵢ|² ≤ ‖x‖². The inequality is a direct consequence of non-negativity of norms — not of individual coefficient bounds.

Question 3

Parseval's identity Σᵢ|⟨x, eᵢ⟩|² = ‖x‖² holding for all x in a Hilbert space is equivalent to the orthonormal system {eᵢ} being a complete orthonormal basis.

• T. True
• F. False

Answer: True

Parseval's identity (equality) holds if and only if the Fourier series converges to x itself, which happens if and only if there is no nonzero vector orthogonal to all eᵢ — the definition of completeness. The identity is literally the Pythagorean theorem extended to infinite dimensions, valid only when all directions are accounted for.

Question 4

If an orthonormal system {eᵢ} satisfies Bessel's inequality but not Parseval's identity for some vector x, the 'missing energy' can be recovered by simply summing more terms in the same Fourier series.

• T. True
• F. False

Answer: False

The missing energy corresponds to the component of x lying in the orthogonal complement of the closed subspace spanned by all the eᵢ. No additional terms drawn from the same system can capture it — those directions simply do not exist in the span. Recovering the energy requires either adding new basis vectors that span the missing directions or recognizing the system is incomplete.

Question 5

An orthonormal system {eᵢ} spans a proper closed subspace V of a Hilbert space H. Explain why the Fourier series Σᵢ⟨x, eᵢ⟩eᵢ converges for any x ∈ H, yet fails to converge to x when x ∉ V.

Model answer: The series converges because Bessel's inequality guarantees Σ|⟨x, eᵢ⟩|² ≤ ‖x‖² < ∞, and in a Hilbert space, square-summability of coefficients implies convergence of the series in norm. However, the sum converges to PV(x), the orthogonal projection of x onto V — not to x itself. The difference x − PV(x) is orthogonal to every eᵢ and represents the component of x in the orthogonal complement of V. Only when V = H (the system is complete) does PV(x) = x and Parseval's identity hold.

The key distinction is between convergence of the series (guaranteed by Bessel) and convergence to x (guaranteed only by completeness). The Fourier series always recovers the projection; Parseval's identity is the statement that this projection is the identity map — that V = H.