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