Functional analysis

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   In $\mathbb{R}^2$ with the Euclidean inner product, let $x = (2, 5)$ and $y = (5, 2)$. The Cauchy–Schwarz inequality states $\langle x, y \rangle^2 \le \lVert x \rVert^2 \, \lVert y \rVert^2$. Compute the value of the right-hand side $\lVert x \rVert^2 \, \lVert y \rVert^2$.

2. Warm-up

   In $\mathbb{R}^2$ with the Euclidean inner product, let $x = (3, 5)$ and $y = (4, 5)$. The Cauchy–Schwarz inequality states $\langle x, y \rangle^2 \le \lVert x \rVert^2 \, \lVert y \rVert^2$. Compute the value of the right-hand side $\lVert x \rVert^2 \, \lVert y \rVert^2$.

3. Warm-up

   In $\mathbb{R}^2$ with the Euclidean inner product, let $x = (1, 5)$ and $y = (1, 4)$. The Cauchy–Schwarz inequality states $\langle x, y \rangle^2 \le \lVert x \rVert^2 \, \lVert y \rVert^2$. Compute the value of the right-hand side $\lVert x \rVert^2 \, \lVert y \rVert^2$.

Problems shown under their original terms: Generated (template, solver-computed).