Questions: Orthogonality in Hilbert Spaces

Question 1

Let M be a closed subspace of a Hilbert space H, and let h ∈ H. The orthogonal projection of h onto M is best described as:

• A. The unique element of M with the smallest norm
• B. The unique element m ∈ M that minimizes ‖h − m‖ — the closest point in M to h
• C. The element of M⊥ that, when added to h, gives zero
• D. Any element of M that is orthogonal to h

Answer: (B)

The orthogonal decomposition theorem guarantees that h = m + m⊥ uniquely, where m ∈ M and m⊥ ∈ M⊥. The element m is the orthogonal projection of h onto M, and it is the closest point in M to h: for any other element n ∈ M, ‖h − m‖ ≤ ‖h − n‖. This best-approximation property is the key geometric content. In L²[−π,π], partial sums of Fourier series are projections onto finite-dimensional subspaces, giving the best approximations to a function by trigonometric polynomials of degree ≤ N.

Question 2

A student argues: 'Orthogonal decomposition should work for any subspace M, not just closed ones — if M has an orthogonal complement, H = M ⊕ M⊥ should hold.' What is the flaw?

• A. Non-closed subspaces have no orthogonal complement at all
• B. M⊥ is well-defined for any subspace, but the closest point to h in M may not exist when M is not closed — closure is needed to guarantee the infimum is achieved
• C. Orthogonal decomposition works for non-closed subspaces in finite dimensions but fails only in infinite dimensions
• D. The student is correct — the theorem holds for any subspace, closed or not

Answer: (B)

M⊥ is always well-defined and closed (for any M), but the problem is existence of the closest point. The infimum of {‖h − m‖ : m ∈ M} is always finite, but may not be achieved — there may be no element in M that actually attains the minimum distance — if M is not closed. In infinite-dimensional spaces, a non-closed subspace may contain sequences converging to elements outside M, so the infimum-achieving limit escapes M. Closure ensures the minimizer exists within M.

Question 3

M⊥ is always a closed subspace of H, even if M itself is not closed.

• T. True
• F. False

Answer: True

M⊥ = {v ∈ H : ⟨v, m⟩ = 0 for all m ∈ M} is an intersection of sets of the form {v : ⟨v, m⟩ = 0}, each of which is closed (the kernel of the continuous linear map v ↦ ⟨v, m⟩). An intersection of closed sets is closed, so M⊥ is always closed regardless of whether M is. The closure of M⊥ follows from continuity of the inner product, not from any property of M itself.

Question 4

If M is a closed subspace of a Hilbert space H, then M ∩ M⊥ can contain non-zero elements, since some vectors might project non-trivially onto both M and its complement.

• T. True
• F. False

Answer: False

M ∩ M⊥ = {0}. If v ∈ M ∩ M⊥, then since v ∈ M⊥, we have ⟨v, m⟩ = 0 for all m ∈ M. But v ∈ M, so in particular ⟨v, v⟩ = 0, which gives ‖v‖² = 0, hence v = 0. The only vector orthogonal to itself is the zero vector. This trivial intersection is why the decomposition h = m + m⊥ is unique and the sum is a true direct sum.

Question 5

How do Fourier series illustrate the orthogonal decomposition theorem in L²[−π,π]? Identify the subspace, the inner product, and what the theorem guarantees.

Model answer: In H = L²[−π,π], the inner product is ⟨f, g⟩ = ∫f(x)g(x)dx. The trigonometric functions {1, cos(x), sin(x), cos(2x), sin(2x), …} form an orthonormal system — each pair has inner product zero, each has unit norm. Each finite collection of these functions spans a closed subspace (a space of trigonometric polynomials). The orthogonal decomposition theorem, extended to the full orthonormal basis, guarantees that any f ∈ L²[−π,π] decomposes uniquely as a Fourier series, and each Fourier coefficient is computed as the inner product ⟨f, eₙ⟩.

Fourier coefficients are inner products: aₙ = (1/π)⟨f, cos(nx)⟩, bₙ = (1/π)⟨f, sin(nx)⟩. The partial sums of the Fourier series are orthogonal projections onto finite-dimensional subspaces, giving the best L² approximation to f by trigonometric polynomials of degree ≤ N. The full theorem guarantees that the series converges in L² norm to f itself. This makes abstract Hilbert space geometry the engine behind one of the most widely used tools in mathematics, physics, and engineering.