List of top Functional Analysis Questions on Linear Operators and Isometries

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 \( a>0 \). 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?