Let ${ (X, d) }$ be a metric space. There is a distance preserving embedding ${ X \hookrightarrow V }$ into a normed space ${ V .}$ Consider ${ \mathcal{B}(X, \mathbb{R}) },$ the space of bounded functions on set ${ X },$ with sup norm. Fix ${ a \in X }.$ Now the map ${ X \hookrightarrow \mathcal{B}(X, \mathbb{R}) }$ sending ${ x \mapsto \varp... Read more 28 Nov 2021 - 1 minute read
[Ref: Prof S. G. Johnson’s proof] Let ${ V }$ be an ${ \mathbb{R}- }$vector space with basis ${ \mathcal{B} = (v _1, \ldots, v _n) },$ and let ${ \lVert \ldots \rVert _{1}, \lVert \ldots \rVert _{2} }$ be two norms on it. Turns out these norms are equivalent. If ${ \mathscr{V} \overset{T}{\to} \mathscr{W} }$ is an isomorphism of ${\mathbb{R}-}... Read more 26 Nov 2021 - 3 minute read
Take an interval of the form ${ [\delta, 2A-\delta] \subseteq \mathbb{R} _{\gt 0} }.$ So here ${ 0 \lt \delta \lt A }.$ Consider ${ \log (x) }$ on ${ [\delta, 2A-\delta] }.$ We’ll find a sequence ${ (P _n (x)) }$ of polynomials of degrees ${ \deg(P _n) \leq n }$ such that ${ \max _{x \in [\delta, 2A - \delta]} \vert \log(x) - P _n (x) \vert }$... Read more 22 Nov 2021 - 2 minute read
[Minor observation] Notation: Let ${ (x _j) _{j \geq 1} }$ be a sequence in a normed space. A sequence ${ (x _j) _{ j \in J} }$ with ${ J \subseteq \mathbb{Z} _{\gt 0} }$ infinite is called a subsequence. We say subsequence ${ (x _j) _{j \in J} }$ converges to ${ p }$ as ${ j \in J, j \to \infty }$ if for every ${ \epsilon \gt 0 },$ ${ x _j ... Read more 11 Nov 2021 - 3 minute read
[An old proof. “By finding scaled versions of the triangle within itself”. Yet somewhat mysterious.] So ${c = ( \frac{b}{c} ) b + ( \frac{a}{c} ) a. }$ Read more 09 Oct 2021 - less than 1 minute read