https://www.chenk.top/en/tags/hilbert-spaces/Recent content in Hilbert-Spaces on Chen Kai BlogHugoenTue, 05 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/03-hilbert-spaces/Tue, 05 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/03-hilbert-spaces/<h2 id="inner-products-and-the-geometry-they-create" class="heading-anchor">Inner Products and the Geometry They Create<a href="#inner-products-and-the-geometry-they-create" 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/03_inner_product.png" alt="Inner product geometry: angle and projection" loading="lazy" decoding="async" class="content-image"> </figure> </p> <p>If a Banach space is a normed space that has agreed to be complete, a Hilbert space is a Banach space that has further agreed to admit angles. That extra agreement — an inner product — is what restores almost all of finite-dimensional geometry to the infinite-dimensional setting. Orthogonality, projection, the Pythagorean theorem, the notion of &ldquo;closest point in a subspace&rdquo; — all come back unchanged. The price of admission is a single axiom; the reward, geometric and computational, is enormous.</p>