Sine and cosine are orthogonal on [0,π] and both have squared norm π/2. In the normalized basis of their span, T has matrix $\begin{pmatrix}0&\pi/2\\\pi/2&0\end{pmatrix}$; it vanishes on the orthogonal complement. Its norm is π/2, so **(4/π)‖T‖ = 2**.
More Hilbert Spaces questions
GATE 2025 MA Q54– An infinite orthonormal set x k lies in a real Hilbert space X. Let Y=spanx₁,…,x n and S…
GATE 2024 MA Q48– In real ℓ²(ℤ) with usual inner product, T sends x to y with y j=(x j+x (−j))/2. Which…
GATE 2024 MA Q57– In real L²[−1,1], ‖f‖²=18/5, ∫f=2 and ∫xf=0. P₁ is the subspace of polynomials of degree…
GATE 2023 MA Q32– On real L²[−1,1], F(f)=∫₋₁¹f(x)x²dx. With the usual L² norm, find 5‖F‖².
GATE 2023 MA Q56– In real ℓ² let M=\x: n1x n/4n=0\. The orthogonal projection of (1,1/2,1/3,…) onto M⊥ is (…
GATE 2022 MA Q16– 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.…