Questions: Bessel's Inequality and Parseval's Identity

Question 1

You have a Hilbert space H and an orthonormal sequence (eₙ). For a specific vector x, you compute Σ|⟨x, eₙ⟩|² and find the sum is strictly less than ‖x‖². What does this tell you?

• A. The computation must contain an error — Parseval's identity requires the sum to equal ‖x‖² for any orthonormal sequence
• B. The vector x has a component that is not captured by the sequence (eₙ) — the sequence is not a complete orthonormal basis
• C. The vector x does not belong to the Hilbert space H
• D. The sequence (eₙ) is not orthonormal — only normalized sequences satisfy the inequality

Answer: (B)

Bessel's inequality guarantees Σ|⟨x, eₙ⟩|² ≤ ‖x‖² for any orthonormal sequence; a strict inequality is perfectly consistent and means the sequence is incomplete — it fails to span a dense subspace of H. The 'missing' energy, ‖x‖² − Σ|⟨x, eₙ⟩|², is the squared norm of the component of x that is orthogonal to every eₙ. Parseval's equality holds only when no such orthogonal component can exist, which is exactly the definition of a complete orthonormal basis.

Question 2

In signal processing, a signal x is analyzed using a complete orthonormal frequency basis. Parseval's identity holds. Which statement correctly interprets this?

• A. The signal has an equal number of time samples and frequency components
• B. The total energy computed from the time-domain signal equals the total energy computed from the Fourier coefficients — no energy is lost in the frequency representation
• C. The signal can be perfectly reconstructed from any finite subset of its Fourier coefficients
• D. The Fourier coefficients all have the same magnitude, since energy is conserved

Answer: (B)

Parseval's identity says ‖x‖² = Σ|⟨x, eₙ⟩|² — the norm (energy) computed in the original domain equals the sum of squared coefficients in the frequency domain. This is an energy conservation statement: the change of basis from time domain to frequency domain preserves total energy. It does NOT mean all coefficients are equal (option D) or that finite truncations suffice (option C). Option C is false because a finite partial sum always loses the energy in the omitted components.

Question 3

Bessel's inequality Σ|⟨x, eₙ⟩|² ≤ ‖x‖² holds for any orthonormal sequence (eₙ) in a Hilbert space, regardless of whether that sequence forms a complete basis.

• T. True
• F. False

Answer: True

Bessel's inequality is unconditional — it follows from the non-negativity of ‖x − Sₙ‖² ≥ 0 for the partial projection Sₙ, and this holds for any orthonormal sequence regardless of completeness. Completeness is the additional condition that makes the inequality into an equality (Parseval). Without completeness, the sum converges to some value ≤ ‖x‖², and the gap represents the squared norm of the projection of x onto the orthogonal complement of the sequence's closed span.

Question 4

If Parseval's identity Σ|⟨x, eₙ⟩|² = ‖x‖² holds for one specific vector x, the orthonormal sequence should be a complete orthonormal basis.

• T. True
• F. False

Answer: False

Parseval holding for a single vector does not imply completeness. A sequence could capture all the energy of one particular vector while failing to span the full space. For example, if x happens to lie in the closed span of the sequence, Parseval holds for x even if the sequence misses an orthogonal subspace entirely. Completeness requires Parseval's identity to hold for ALL vectors in H — it is a global condition on the sequence, not a local condition on any single vector.

Question 5

What is the precise condition that distinguishes Parseval's identity (equality) from Bessel's inequality, and what does this condition mean geometrically in the Hilbert space?

Model answer: The condition is completeness: the orthonormal sequence (eₙ) must form a complete orthonormal basis, meaning its closed linear span is all of H (or equivalently, the only vector orthogonal to every eₙ is the zero vector). Geometrically, this means no vector in H has a nonzero 'shadow' outside the span of the basis. When the basis is complete, the partial projections Sₙ = Σᵢ⁼¹ⁿ⟨x,eᵢ⟩eᵢ converge in norm to x itself (‖x − Sₙ‖ → 0), so all of ‖x‖² is accounted for by the coefficients and none is lost to an orthogonal complement.

Bessel's inequality always holds because the projection onto any finite-dimensional subspace never exceeds the original norm. The inequality becomes equality exactly when the 'leftover' ‖x − Sₙ‖ → 0, which happens if and only if the basis spans a dense subspace. The concept of completeness in infinite-dimensional Hilbert spaces is subtler than in finite dimensions (where any basis automatically spans the space), making Parseval the canonical criterion for testing whether an orthonormal sequence is truly a basis.