List of top Functional Analysis Questions on Functional Analysis

Let \( T : L^2[-1, 1] \to L^2[-1, 1] \) be defined by \( T f = \tilde{f} \), where \( \tilde{f}(x) = f(-x) \) almost everywhere. If \( M \) is the kernel of \( I - T \), then the distance between the function \( \varphi(t) = e^t \) and \( M \) is

Let \( H \) be a complex Hilbert space. Let \( u, v \in H \) be such that \( \langle u, v \rangle = 2 \). Then \[ \frac{1}{2\pi} \int_0^{2\pi} \| u + e^{it} v \|^2 e^{it} dt = \underline{\hspace{1cm}}. \]

Let \( \ell^1 = \{ x = (x(1), x(2), \dots, x(n), \dots) : \sum_{n=1}^{\infty} |x(n)| < \infty \} \) be the sequence space equipped with the norm \( \|x\| = \sum_{n=1}^{\infty} |x(n)| \). Consider the subspace \[ X = \left\{ x \in \ell^1 : \sum_{n=1}^{\infty} |x(n)| < \infty \right\}, \] and the linear transformation \( T: X \to \ell^1 \) given by \( (Tx)(n) = n x(n) \text{  for  } n = 1, 2, 3, \dots. \) Then:

Let \( C[0, 1] \) be the Banach space of real valued continuous functions on [0, 1] equipped with the supremum norm. Define \( T: C[0, 1] \to C[0, 1] \) by \[ (Tf)(x) = \int_0^x t f(t) \, dt. \] Let \( R(T) \) denote the range space of \( T \). Consider the following statements: \[\begin{array}{rl} \bullet & \text{P: \( T \) is a bounded linear operator.} \\ \bullet & \text{Q: \( T^{-1}: R(T) \to C[0, 1] \) exists and is bounded.} \\ \end{array}\]