GATE Functional Analysis: Bounded Linear Operators and Duality – Previous Year Questions
15 GATE previous year questions on Bounded Linear Operators and Duality (Functional Analysis, Mathematics) with answers and explanations, from every paper.
- GATE 2021 MA Q19 – For an orthonormal basis (e n) of a complex Hilbert space H: P:there is a bounded linear f:H→ℂ with f(e n)=1/n. Q:there is a bounded linear g:H→ℂ with…
- GATE 2021 MA Q44 – On C[0,1] with supremum norm define (Tf)(x)=∫₀ˣx f(t)dt. Let R(T) be its range. P:T is bounded linear. Q:T⁻¹:R(T)→C[0,1] exists and is bounded. Choose…
- GATE 2021 MA Q45 – Let ℓ¹ have its usual norm and X=x∈ℓ¹:Σₙ₌₁∞n x(n) <∞. Define T:X→ℓ¹ by (Tx)(n)=nx(n), using on X the inherited ℓ¹ norm. Which holds?
- GATE 2022 MA Q43 – X,Y,Z are Banach spaces, T:X→Y linear, and S:Y→Z bounded linear and injective. If S∘T is bounded, which must hold?
- GATE 2022 MA Q51 – For a>0 define D a:L²(ℝ)→L²(ℝ) by (D af)(x)=a(−1/2)f(x/a) almost everywhere. Which hold?
- GATE 2023 MA Q23 – In real C[0,1] with supremum norm let X=f:f(0)=f(1)=0, S X=f∈X:‖f‖∞=1 and F(f)=∫₀¹f(t)dt. How many elements are in f∈X:F(f)=‖F‖∩S X?
- GATE 2023 MA Q37 – On real ℓ² with usual norm, S(x₁,x₂,…)=(0,x₁,x₂,…) and T(x₁,x₂,…)=(0,x₂,0,x₄,…). Which holds?
- GATE 2023 MA Q38 – Let c₀₀ be finitely supported real sequences with supremum norm. Define F(x) n=(n+1/n)x n and G(x) n=x n/(n+1/n). Which holds?
- GATE 2024 MA Q47 – On finitely supported real sequences C₀₀ define f(x)=Σₙ₌₁∞x n/√n. I:f extends continuously linearly to ℓ¹. II:f extends continuously linearly to ℓ².…
- GATE 2024 MA Q49 – X=ℝ² with norm x + y . On S=(x,0), f(x,0)=2x. A Hahn–Banach extension F agrees with f and preserves its norm. Which statements hold?
- GATE 2025 MA Q17 – X=f∈C[0,1]:f(0)=f(1)=0 with sup norm; Y=C[0,1] with L² norm. On their closed unit balls U X,U Y, define Tf=Sf=∫₀¹f(t)dt. S1: sup over U X of Tf is…
- GATE 2025 MA Q50 – For 0<α<1, Cα[0,1] is the Hölder Banach space with norm f(0) +sup (s≠t) f(t)−f(s) / t−s α. T:Cα→C[0,1] is inclusion into sup norm. Which statements…
- GATE 2026 MA Q32 – On ℝ³ with ℓ¹ norm, T has matrix pmatrix1&1&3\\2&2&-2\\1&3&-3pmatrix. Find its operator norm (integer).
- GATE 2026 MA Q42 – On ℓ∞ with supremum norm, T(x 1,x 2,)=(x 1,x 2/2,x 3/3,). Which statement holds?
- GATE 2026 MA Q53 – For a normed linear space X and its continuous dual X′, which statements hold?