GATE Functional Analysis: Hilbert Spaces – Previous Year Questions

10 GATE previous year questions on Hilbert Spaces (Functional Analysis, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q35 (1 mark, Numerical answer) – In a complex Hilbert space, ⟨u,v⟩=2. Evaluate (2)-1 02\ u+eitv\ 2eitdt.
2. GATE 2021 MA Q43 (2 marks, Multiple choice) – In real L²[−1,1] with its usual norm, let M=f:∫₋₁¹f=0 and f(x)=x². Find inf (g∈M)‖f−g‖.
3. GATE 2022 MA Q16 (1 mark, Multiple choice) – Let (e k) be an orthonormal Hilbert-space basis, f k=e k+e (k+1), and g j= n=1j(-1)n+1e n. Find k=1 g j,f k 2.
4. GATE 2022 MA Q42 (2 marks, Multiple choice) – On L²[−1,1] define (Tf)(x)=f(−x), and let M=ker(I−T). Find the distance from φ(t)=eᵗ to M.
5. GATE 2023 MA Q32 (1 mark, Numerical answer) – On real L²[−1,1], F(f)=∫₋₁¹f(x)x²dx. With the usual L² norm, find 5‖F‖².
6. GATE 2023 MA Q56 (2 marks, Numerical answer) – In real ℓ² let M=\x: n1x n/4n=0\. The orthogonal projection of (1,1/2,1/3,…) onto M⊥ is ( n11/(n4n))(1/4,1/42,1/43,). Find α.
7. GATE 2024 MA Q48 (2 marks, Multiple select) – In real ℓ²(ℤ) with usual inner product, T sends x to y with y j=(x j+x (−j))/2. Which statements hold?
8. GATE 2024 MA Q57 (2 marks, Numerical answer) – In real L²[−1,1], ‖f‖²=18/5, ∫f=2 and ∫xf=0. P₁ is the subspace of polynomials of degree at most one. Find inf (g∈P₁)‖f−g‖² to two decimals.
9. GATE 2025 MA Q54 (2 marks, Multiple select) – An infinite orthonormal set x k lies in a real Hilbert space X. Let Y=spanx₁,…,x n and S nx= k=1n x,x k x k. Which statements hold?
10. GATE 2026 MA Q60 (2 marks, Numerical answer) – On real L²[0,π], Tf(x)= x 0 f(t) t\,dt+ x 0 f(t) t\,dt. Find (4/π)‖T‖ (integer).