$\DeclareMathOperator{\bo}{\small Bo}$ Lectures 1-6
Let $X, Y$ and $Z$ be normed vector spaces, and let $ℒ(X, Y)$ denote the space of linear maps $X→Y$ and $ℬ(X, Y)$ the subset of bounded linear maps.
Let $X$ and $Y$ be finite-dimensional normed vector spaces, and equip $X⊕Y$ with the norm ${‖(x, y)‖}={‖x‖}+{‖y‖}$. A map $β: X×Y→ℝ$ is said to be bilinear if it is linear in each factor.
$X×Y$ can be viewed as a normed vector space, where the vector space structure is given componentwise (i.e. $X⊕Y$) and the norm is given by setting ${‖(x, y)‖}={‖x‖}+{‖y‖}$ (for all $x∈X, y∈Y$). Show that $β: X⊕Y→ℝ$ is differentiable and calculate its derivative at a point $(x, y)∈X⊕Y$.
Let $x_1,…,x_n$ be a basis of $X$, $y_1,…,y_m$ be a basis of $Y$. Let $M=\max_{\substack{1≤i≤n\\1≤j≤m}} β(x_i,y_j)$.
For $a=a_1x_1+⋯+a_nx_n,b=b_1y_1+⋯+b_my_m$, \[β(a,b)=\sum_{i=1}^n\sum_{j=1}^m{|a_ib_j|}β(x_i,y_j)≤M\sum_{i=1}^n\sum_{j=1}^m{|a_ib_j|}=M{|a|}{|b|}≤M({|a|}+{|b|})^2\] So \[\lim_{(a,b)→0}{|β(a,b)|\over{|a|}+{|b|}}≤\lim_{(a,b)→0}{M({|a|}+{|b|})^2\over{|a|}+{|b|}}=0\] To prove $Df(a,b)(x,y)=β(a,y)+β(x,b)$ \begin{align*} \lim_{(a,b)→0}{|β(x+a,y+b)-β(x,y)-β(a,y)-β(x,b)|\over{|a|}+{|b|}}&=\lim_{(a,b)→0}{|β(a,b)|\over{|a|}+{|b|}}=0 \end{align*}
Let $X$ and $Y$ be finite-dimensional normed vector spaces and let $U⊆X$ be an open subset.
Suppose $U$ is an open subset of ℝ and $f: U→ℝ$ is differentiable on $U$.
Show that $f∈𝒞^1$ if and only if $f$ is strongly continuous i.e. the function $s: U×U→ℝ$ $$s(x, y)=\left\{\begin{array}{cc} \frac{f(x)-f(y)}{x-y}, & x≠y, \\ f'(x), & x=y . \end{array}\right. $$is continuous.
⇐ Suppose $s$ is continuous. $f'$ is the composition of $s$ and $x↦(x,x)$, so $f'$ is continuous, so $f∈𝒞^1$.
⇒ Suppose $f'$ is continuous. Then $s$ is continuous on $U^2∖\{(x,y):x=y\}$.
$s$ is continuous at $(a,a)∈U^2$ iff $∀ϵ>0∃δ:f(B((a,a),δ))⊂B((a,a),ϵ)$, wlog we use 1-norm in $U^2$.
By Mean Value Theorem $∃ξ ∈ ℝ$ between $x,y$ such that $s(x,y)=f'(ξ)$
By uniform continuity of $f'$, $∃δ$ such that $∀z∈B(a,δ):{|f'(z)-f'(a)|}≤ϵ.$
\[{‖(x,y)-(a,a)‖}≤δ⟹{|x-a|}+{|y-a|}≤δ⟹{|ξ-a|}≤δ⟹{|s(x,y)-s(a,a)|}={|f'(ξ)-f'(a)|}≤ϵ\]
Let $ι: \mathrm{GL}_n(ℝ)→\mathrm{GL}_n(ℝ)$ denote the inversion map, and view $\mathrm{GL}_n(ℝ)$ as an open subset of $\mathrm{Mat}_n(ℝ)$ where the latter is a normed vector space (via either the operator or Hilbert-Schmidt norm).