https://www.chenk.top/en/tags/normed-spaces/Recent content in Normed-Spaces on Chen Kai BlogHugoenSun, 03 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/02-normed-and-banach/Sun, 03 Oct 2021 09:00:00 +0000https://www.chenk.top/en/functional-analysis/02-normed-and-banach/<h2 id="why-a-norm-is-more-than-a-metric-wearing-a-hat" class="heading-anchor">Why a Norm Is More Than a Metric Wearing a Hat<a href="#why-a-norm-is-more-than-a-metric-wearing-a-hat" 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/02_lp_morph.gif" alt="Animation: l^p unit ball morphing as p changes" loading="lazy" decoding="async" class="content-image"> </figure> </p> <p>In Article 1, the metric was a free-standing function on a set with no algebraic structure. That generality bought us topology and completeness, but it gave nothing back to the algebra. The moment I am willing to assume the underlying set is a vector space, a more rigid object becomes available: a <strong>norm</strong>, a single nonnegative function on the space whose induced metric <span class="math-inline">$d(x,y) = \|x - y\|$</span> is <em>translation-invariant</em> and <em>positively homogeneous</em>.</p>