https://www.chenk.top/en/tags/weak-topology/Recent content in Weak-Topology on Chen Kai BlogHugoenSat, 09 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/05-weak-topologies/Sat, 09 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/05-weak-topologies/<h2 id="why-weaker-topologies-exist-and-why-they-matter" class="heading-anchor">Why Weaker Topologies Exist and Why They Matter<a href="#why-weaker-topologies-exist-and-why-they-matter" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/functional-analysis/figures/fa05_weak_convergence.png" alt="Strong convergence vs weak convergence" loading="lazy" decoding="async" class="content-image"> </figure> </p> <p>Article 1 ended with a depressing fact: in any infinite-dimensional normed space, the closed unit ball is not compact. No bounded sequence is guaranteed to have a norm-convergent subsequence. If you are trying to find a minimizer of an energy functional — say, the lowest-energy configuration of a vibrating membrane — you take a minimizing sequence, and you need a limit. In finite dimensions, Bolzano-Weierstrass delivers that limit. In infinite dimensions, it does not. The direct method of the calculus of variations appears dead on arrival.</p>