Th: Consider a sequence ${ (a _n) \subseteq \mathbb{R} }.$ Let ${ \mathbb{R} _{\gt 0} \overset{f}{\to} \mathbb{R} }$ be a ${ \mathcal{C} ^{1} }$ function, and ${ (x _n) \subseteq \mathbb{R} _{\gt 0} }$ a seq with ${ x _n \nearrow \infty }.$ Using ${ (x _n)}$ for indexing gives functions ${ S(X) := \sum _{x _n \leq X} a _n }$ and ${ S _f (X) := ... Read more 19 Dec 2021 - 4 minute read
(Ref: Kreyszig’s “Functional Analysis”; P. Tamaroff’s post) Let ${ (X,d) }$ be a metric space. A metric space ${ (\hat{X}, \hat{d}) }$ with the properties ${ (\hat{X}, \hat{d}) }$ is complete There is an embedding ${ (X,d) \overset{T}{\hookrightarrow} (\hat{X}, \hat{d})}$ Every ${ p \in \hat{X} }$ is the limit of some Cauchy seq containe... Read more 14 Dec 2021 - 2 minute read
Th: Let ${ \mathbb{F} }$ be ${ \mathbb{R} }$ or ${ \mathbb{C} }.$ Let ${ V }$ be a normed ${ \mathbb{F}- }$vector space with a basis ${ \mathcal{B} = (b _1, \ldots, b _k) }.$ Now ${ V }$ is complete. Pf: We have an isomorphism ${ \mathbb{F} ^k \to V }$ sending ${ x = (x _1 , \ldots, x _k ) ^{t} \mapsto \mathcal{B}x = \sum _{1} ^{k} x _j b _j .}... Read more 03 Dec 2021 - 2 minute read
Consider ${ \mathcal{C}([0,1]) },$ the space of continuous functions on ${ [0,1] }.$ For ${ p \geq 1 }$ it has ${ p- }$norm ${ \lVert f \rVert _{p} := (\int \vert f \vert ^p ) ^{\frac{1}{p}} }.$ It also has sup norm ${ \lVert f \rVert _{\infty} := \sup _{t \in [0,1]} \vert f(t) \vert }.$ For now we’ll only consider ${ \lVert .. \rVert _{1}, }$ ... Read more 29 Nov 2021 - 6 minute read
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