List of top Functional Analysis Questions on Hilbert Spaces asked in GATE MA

Let \( L^2[0, \pi] \) denote the space of all real valued Lebesgue square integrable functions on \( [0, \pi] \). Let \( T: L^2[0, \pi] \to L^2[0, \pi] \) be defined as follows: \[ T(f(x)) = \sin x \int_0^{\pi} f(t)\cos t \, dt + \cos x \int_0^{\pi} f(t)\sin t \, dt \]

Then the value of \( \dfrac{4}{\pi}\|T\| \) is equal to ______. (Answer in integer)

Let \( L^2[-1, 1] \) be the Hilbert space of real-valued square integrable functions on [-1, 1] equipped with the norm \[ \|f\| = \left( \int_{-1}^{1} |f(x)|^2 \, dx \right)^{1/2}. \] Consider the subspace \[ M = \{ f \in L^2[-1, 1] : \int_{-1}^{1} f(x) \, dx = 0 \}. \] For \( f(x) = x^2 \), define \[ d = \inf \{ \|f - g\| : g \in M \}. \] Then

Let \( \{ e_n : n = 1, 2, 3, \dots \} \) be an orthonormal basis of a complex Hilbert space \( H \). Consider the following statements: P: There exists a bounded linear functional \( f: H \to \mathbb{C} \) such that \( f(e_n) = \frac{1}{n} \) for \( n = 1, 2, 3, \dots \)
Q: There exists a bounded linear functional \( g: H \to \mathbb{C} \) such that \( g(e_n) = \frac{1}{\sqrt{n}} \) for \( n = 1, 2, 3, \dots \)