MCQs on Hilbert Spaces

Which of the following is a complete inner product space?

• Normed space

• Banach space

• Hilbert space

• Metric space

The inner product ⟨x, y⟩ in a Hilbert space is:

• Linear in both variables

• Linear in the first, conjugate linear in the second

• Always symmetric

• Defined only for real vector spaces

In a real Hilbert space, the inner product satisfies:

• ⟨x, y⟩ = −⟨y, x⟩

• ⟨x, y⟩ = ⟨y, x⟩

• ⟨x, y⟩ = ⟨x, x⟩

• ⟨x, y⟩ = 0 for all x, y

The norm induced by an inner product is given by:

• ‖x‖ = ⟨x, x⟩

• ‖x‖ = √x

• ‖x‖ = √⟨x, x⟩

• ‖x‖ = |x|

Which inequality holds in every inner product space?

• Triangle inequality

• Holder's inequality

• Cauchy-Schwarz inequality

• Minkowski inequality

A set {e₁, e₂, …} in a Hilbert space is an orthonormal set if:

• ⟨eᵢ, eⱼ⟩ = 1

• eᵢ = eⱼ for all i ≠ j

• ⟨eᵢ, eⱼ⟩ = δᵢⱼ

• ‖eᵢ‖ = 0

The best approximation of a vector x in a closed subspace M of a Hilbert space H is:

• Any vector in M

• A vector orthogonal to x

• The orthogonal projection of x onto M

• The zero vector

A Hilbert space is separable if:

• It has only one element

• It has a countable dense subset

• It is finite-dimensional

• Every subset is compact

The Fourier series expansion in a Hilbert space converges:

• Uniformly

• Pointwise

• In norm (mean-square sense)

• Almost everywhere

If {eₙ} is an orthonormal basis of H and x ∈ H, then:

• x = Σ⟨x, eₙ⟩eₙ

• x = Σeₙ

• x = ⟨x, x⟩

• x = 0