Let ${ f \in \mathcal{C}[a,b] }.$ Weierstrass approximation theorem says there is a sequence of polynomials uniformly converging to ${ f }$ on ${ [a,b] }.$ That is, for every ${ \epsilon \gt 0 }$ there is a polynomial ${ P }$ with ${ \max _{x \in [a,b]} \vert f (x) - P(x) \vert }$ ${ \lt \epsilon }.$ The proof below is originally due to Lebesg... Read more 22 Jan 2022 - 7 minute read
(Ref: Stromberg’s “Classical Real Analysis”) Let ${ \alpha \in \mathbb{R} }.$ The generalised binomial theorem says ${ (1+x) ^{\alpha} }$ ${ = \sum _{0} ^{\infty} \binom{\alpha}{j} x ^j }$ for ${ \vert x \vert \lt 1 }.$ Also, radius of convergence of ${ \sum _{1} ^{\infty} \binom{\alpha}{j} x ^j }$ is ${ \infty }$ when ${ \alpha \in \mathbb{... Read more 15 Jan 2022 - 4 minute read
Let ${ F }$ be a normed space and ${ X }$ a set. For a map ${ X \overset{f}{\to} F },$ let ${ \lVert f \rVert _{\infty} }$ ${ := \sup _{ x \in X} \vert f (x) \vert }$ ${ \underline{ \in [0, \infty] } }.$ ${ \lVert \ldots \rVert _{\infty}, }$ when restricted to the space ${ \mathcal{B}(X, F) }$ of bounded functions from ${ X }$ to ${ F, }$ be... Read more 14 Jan 2022 - 15 minute read
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