List of top Functional Analysis Questions

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 \(X\) be any normed linear space and \(X'\) be the dual space of \(X\). Which of the following statements is/are TRUE?

Let \(\ell^{\infty} = \{x = (x_n)_{n \geq 1} \mid x_n \in \mathbb{R},\ \sup\{|x_n| : n = 1, 2, \ldots\} < \infty\}\) with the supremum norm. Let \(T : \ell^{\infty} \to \ell^{\infty}\) be given by \(T(x_1, x_2, x_3, \ldots) = \left(x_1, \dfrac{x_2}{2}, \dfrac{x_3}{3}, \ldots \right)\). Which one of the following is TRUE?

Let \(E_1\) and \(E_2\) be subsets of a normed linear space \(X\), and
\[ E_1 + E_2 = \{x+y : x \in E_1,\ y \in E_2\} \] \[ E_1 \times E_2 = \{(x,y) : x \in E_1,\ y \in E_2\}. \] Which one of the following is NOT TRUE?

Let \(X = (\mathbb{R}^3, \|\cdot\|_1)\), where \(\left\|\begin{pmatrix} x \\ y \\ z \end{pmatrix}\right\|_1 = |x| + |y| + |z|\), and let \(T: X \to X\) be the linear transformation defined by \[ T\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 & 1 & 3 \\ 2 & 2 & -2 \\ 1 & 3 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}. \] The operator norm of \(T\) is equal to ______. (Answer in integer)

Let \( X \) be the space of all continuously differentiable real valued functions on \( [0,1] \). Define the following norms on \( X \):
\[ p_1(x) = \sup\{ |x(t)| : t \in [0,1] \} \]
\[ p_2(x) = \sup\left\{ \left| \frac{d}{dt} x(t) \right| : t \in [0,1] \right\} \]
and \( p_3(x) = p_1(x) + p_2(x) \).
Which one of the following is TRUE?