Basic Analysis I — Real Analysis · Problem 7.1.13

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · metric-spaces

Problem

Consider ℓ2\ell^2 the set of sequences {xn}n=1∞\{x_n\}_{n=1}^\infty of real numbers such that ∑n=1∞xn2<∞\sum_{n=1}^\infty x_n^2 < \infty. a) Prove the Cauchy--Schwarz inequality for two sequences {xn}n=1∞\{x_n\}_{n=1}^\infty and {yn}n=1∞\{y_n\}_{n=1}^\infty in ℓ2\ell^2: Prove that ∑n=1∞xnyn\sum_{n=1}^\infty x_n y_n converges (absolutely) and (∑n=1∞xnyn)2≤(∑n=1∞xn2)(∑n=1∞yn2).\left(\sum_{n=1}^\infty x_n y_n\right)^2 \leq \left(\sum_{n=1}^\infty x_n^2\right)\left(\sum_{n=1}^\infty y_n^2\right). b) Prove that ℓ2\ell^2 is a metric space with the metric d(x,y):−∑n=1∞(xn−yn)2d(x,y) \coloneq \sqrt{\sum_{n=1}^\infty (x_n - y_n)^2}. Hint: Don't forget to show that the series for d(x,y)d(x,y) always converges to some finite number.

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