List of top Functional Analysis Questions

Let \( X \) be a real normed linear space. Let \( X_0 = \{x \in X : \|x\| = 1\} \). If \( X_0 \) contains two distinct points \( x \) and \( y \) and the line segment joining them, then, which of the following statements is TRUE?

Let \( \{ e_k : k \in \mathbb{N} \} \) be an orthonormal basis for a Hilbert space \( H \).
Define \( f_k = e_k + e_{k+1}, k \in \mathbb{N} \) and  \(g_j = \sum_{n=1}^{j} (-1)^{n+1} e_n, j\) \(\in \mathbb{N}.\) 
\(\text{Then}\) \(\quad \sum_{k=1}^{\infty} | \langle g_j, f_k \)\(\rangle |^2 = \, ? \)
 

Let \( \{ \varphi_0, \varphi_1, \varphi_2, \dots \} \) be an orthonormal set in \( L^2[-1, 1] \) such that \( \varphi_n = C_n P_n \), where \( C_n \) is a constant and \( P_n \) is the Legendre polynomial of degree \( n \), for each \( n \in \mathbb{N} \setminus \{0\} \). Then, which of the following statements are TRUE?
Let \( a>0 \). Define \( D_a : L^2(\mathbb{R}) \to L^2(\mathbb{R}) \) by \[ (D_a f)(x) = \frac{1}{\sqrt{a}} f\left( \frac{x}{a} \right), \text{ almost everywhere, for } f \in L^2(\mathbb{R}). \] Then, which of the following statements are TRUE?
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 \( L^2[-1, 1] \) 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 \( C[0, 1] \) 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}\]
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 \( n = 1, 2, 3, \dots \)