So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review:
Given a function $\mathbf{f}:\mathbb{R}^{n}\to\mathbb{R}^m$, we say that $\mathbf{f}$ is differentiable at a point $\mathbf{a}=(a^{1},\ldots,a^{n})\in\mathbb{R}^n$ (considered as a $1\times n$ matrix) if there exists a linear transformation, $D\mathbf{f}(\mathbf{a}):\mathbb{R}^n\to\mathbb{R}^m$, (considered as an $m\times n$ matrix) such that $$\lim\limits_{\mathbf{h}\to\mathbf{a}}\dfrac{||\mathbf{f}(\mathbf{a}+\mathbf{h})-\mathbf{f}(\mathbf{a})-D\mathbf{f}(\mathbf{a})(\mathbf{h})||}{||\mathbf{h}||}=0.$$
$D\mathbf{f}(\mathbf{a})$ is called the total derivative or Jacobian matrix of $\mathbf{f}$ at $\mathbf{a}$ and is unique.
We are then asked to prove that if $\mathbf{f}:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^p$ is bilinear, then $$\lim\limits_{(\mathbf{h},\mathbf{k})\to\mathbf{0}}\dfrac{||\mathbf{f}(\mathbf{h},\mathbf{k})||}{||(\mathbf{h},\mathbf{k})||}=0.$$
The best approach I could think of was that $||\mathbf{f}(\mathbf{h},\mathbf{k})||=||\mathbf{h}||\cdot||\mathbf{k}||\cdot||\mathbf{f}(\hat{\mathbf{h}},\hat{\mathbf{k}})||$, where $\hat{\mathbf{x}}=\frac{\mathbf{x}}{||\mathbf{x}||}$. We then have that $||\mathbf{h}||\cdot||\mathbf{k}||\leq||\mathbf{h}||^{2}+||\mathbf{k}||^{2}=||(\mathbf{h},\mathbf{k})||^{2}$, which gives us $$\dfrac{||\mathbf{f}(\mathbf{h},\mathbf{k})||}{||(\mathbf{h},\mathbf{k})||}\leq||(\mathbf{h},\mathbf{k})||\cdot||\mathbf{f}(\hat{\mathbf{h}},\hat{\mathbf{k}})||,$$ where $||\hat{\mathbf{h}}||=||\hat{\mathbf{k}}||=1$. And since $||\mathbf{f}(\mathbf{x},\mathbf{y})||\leq||\mathbf{f}(\hat{\mathbf{x}},\mathbf{y})||\cdot||\mathbf{x}||$ (similarly for $\mathbf{y}$) then $f$ is a bounded (continuous) linear transformation and
$$||\mathbf{f}||_{x}=\sup\{||\mathbf{f}(\mathbf{x},\mathbf{y})||:||\mathbf{x}||=1\}<\infty$$
$$||\mathbf{f}||_{y}=\sup\{||\mathbf{f}(\mathbf{x},\mathbf{y})||:||\mathbf{y}||=1\}<\infty,$$
which I was hoping would imply that $||\mathbf{f}(\hat{\mathbf{h}},\hat{\mathbf{k}})||<\infty$, and so we'd have the result we're looking for. But I am stuck here.
Assuming this result, I was able to complete the problem and prove that $$D\mathbf{f}(\mathbf{a},\mathbf{b})(\mathbf{x},\mathbf{y})=\mathbf{f}(\mathbf{a},\mathbf{y})+\mathbf{f}(\mathbf{x},\mathbf{b}).$$
The author then proceeds to ask us to prove the following:
Given a multilinear function $\mathbf{f}:\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_k}\to\mathbb{R}^p$, show that for $\mathbf{h}=(\mathbf{h}_1,\dots,\mathbf{h}_k)$, with $\mathbf{h}_i\in\mathbb{R}^{n_i}$, we have
$$\lim\limits_{\mathbf{h}\to\mathbf{0}}\dfrac{||\mathbf{f}(\mathbf{a}_1,\ldots,\mathbf{a}_{i-1},\mathbf{h}_i,\mathbf{a}_{i+1}\ldots,\mathbf{a_{j-1}},\mathbf{h}_j,\mathbf{a}_{j+1},\ldots,\mathbf{a}_k)||}{||\mathbf{h}||}=0,$$ for $i\neq j$. Use this to prove that
$$D\mathbf{f}(\mathbf{a}_1,\ldots,\mathbf{a}_k)(\mathbf{x}_1,\ldots,\mathbf{x}_k)=\sum_{i=1}^{k}{\mathbf{f}(\mathbf{a}_1,\ldots,\mathbf{a}_{i-1},\mathbf{x}_i,\mathbf{a}_{i+1}\ldots,\mathbf{a}_k)}.$$
Using the hint from the book of considering the bilinear function, $\mathbf{g}(\mathbf{x},\mathbf{y})=\mathbf{f}(\mathbf{a}_1,\ldots,\mathbf{x},\ldots,\mathbf{y},\ldots,\mathbf{a}_k)$, I was able to show the first part of the problem assuming the above result (which I still can't prove). However I'm now also stuck on how to prove the rest of the question.
For example, consider the specific case with three arguments. Using the fact that $\mathbf{f}$ is multilinear, we have
$$\mathbf{f}(\mathbf{a}+\mathbf{h_1},\mathbf{b}+\mathbf{h_2},\mathbf{c}+\mathbf{h_3})-\mathbf{f}(\mathbf{a},\mathbf{b},\mathbf{c})=$$
$$\mathbf{f}(\mathbf{a},\mathbf{b},\mathbf{h_3})+\mathbf{f}(\mathbf{a},\mathbf{h_2},\mathbf{c})+\mathbf{f}(\mathbf{h_1},\mathbf{b},\mathbf{c}) +$$
$$+\mathbf{f}(\mathbf{a},\mathbf{h_2},\mathbf{h_3})+\mathbf{f}(\mathbf{h_1},\mathbf{b},\mathbf{h_3})+\mathbf{f}(\mathbf{h_1},\mathbf{h_2},\mathbf{c})+$$
$$+\mathbf{f}(\mathbf{h_1},\mathbf{h_2},\mathbf{h_3}).$$
The first three terms are what should be $D\mathbf{f}(\mathbf{a},\mathbf{b},\mathbf{c})$ while the next three will disappear in the limit given the first part of the proof. The trouble is how do I control the limit $\lim\limits_{\mathbf{h}\to\mathbf{0}}\frac{||\mathbf{f}(\mathbf{h_1},\mathbf{h_2},\mathbf{h_3})||}{||(\mathbf{h_1},\mathbf{h_2},\mathbf{h_3})||}$? This trouble arises in the general case as well, any time the number of $\mathbf{h_i}$ is more than two in any single term. E.g. how would I control the term $\mathbf{f}(\mathbf{a_1},\mathbf{a_2},\mathbf{h_3},\mathbf{h_4})$, which has two "$h$" and "$a$" terms each?
Thanks for any help. My analysis skills are a little rusty as I've been focusing on passing my algebra qual.
Best Answer
this is just a comment to help you to see why $k$-linear functions on finite dimensional spaces are bounded.
to see that bilinear forms in finite dimensional spaces are bounded you can argue like that: (using your notation)
Let ${\bf x}=(x_1,\ldots,x_n)$ and ${\bf y}=(y_1,\ldots,y_m)$ be unit vectors in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, then $$ \left\|{\bf f}({\bf x},{\bf y})\right\|=\left\|\sum_{i=1}^n\sum_{j=1}^m x_iy_j\ {\bf f}(e_i,e_j)\right\| \leq \max_{i,j}\ \|{\bf f}(e_i,e_j)\|\sum_{i=1}^n\sum_{j=1}^m (x^2_i+y^2_j). $$ If we call $M=\max_{i,j}\ \|{\bf f}(e_i,e_j)\|$ we have from the above inequality that $$ \|{\bf f}({\bf x},{\bf y})\| \leq M(m+n)\|{\bf x}\|^2\cdot \|{\bf y}\|^2 $$ Since ${\bf x}$ and ${\bf y}$ are unit vectors follows that ${\bf f}$ is bounded by $M(m+n)$. I hope you can extend this for any $k$-linear function.