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Problem: Let $X$ and $Y$ be two normed linear spaces. A function $ϕ : X×Y → \mathbb{R}$ is bilinear if $f(·, y) ∈ L(X, \mathbb{R})$ and $f(x, ·) ∈ L(Y, \mathbb{R})$ for each $(x, y) ∈ X × Y$ . Show that any continuous bilinear function ϕ is Fréechet differentiable with $D_{ϕ,(x,y)}$ $(z, w)$ = $ϕ(z, y) + ϕ(x, w)$

I have attempted a proof but I require help on the last step.


Proof:

Let $\phi: X$ x Y$ \rightarrow \mathbb{R}$ be bilinear and continuous. Fix $(x,y) \in int(X $x$ Y)$. We have that $\phi$ is Fréchet Differentiable if $\exists D_{\phi,(x,y)}$ such that

$$\lim_{v\to 0} (1/||v||) * [\phi(x+v) - \phi(x) - D_{\phi,(x,y)}(v)] = 0 $$

for some $v \in V$, V a neighborhood of 0 in $X x Y$ such that $x+v \in int(X x Y), \forall v \in V$.

Take x = $(x_1,x_2)$ and v $ = (v_1, v_2)$

By bilinearity, $\phi(x + v)$ = $\phi(x_1, x_2) + \phi(v_1, x_2) + \phi(x, v_2) + \phi(v_1, v_2)$

We then have that

$$\lim_{v\to 0} (1/||v||) * [\phi(v_1, x_2) + \phi(x,v_2) + \phi(v_1, v_2) - D_{\phi,(x,y)}(v)] = 0 $$

A potential candidate for the existence of a Fréchet derivative such that this limit tends to zero is $D_{\phi,(x,y)}(v) = \phi(v_1, y) + \phi(x, v_2)$. If I prove that

$$\lim_{v\to 0} (1/||v||) * \phi(v_1, v_2) = 0 \space (2)$$

Then it means the Fréchet derivative exists and is equal to the one I proposed.


I have stopped here because I couldn't find a way to use continuity to get (2). Anyone have any idea of a norm I could choose or how I could use that? Is everything ok to that point?

Thanks

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  • $\begingroup$ Hint: Since $\phi$ is continuous $\|\phi(v_1, v_2)\|\leq C\|v_1\|\|v_2\|$. $\endgroup$ – PtF Feb 23 '18 at 22:21
  • $\begingroup$ Thank you @PtF. That is a result of bilinearity, right? Because my professor hasn't exposed that to us and without it's hard to reach the result... Noting more, it sounds like a C-lipschitz function $\endgroup$ – jpugliese Feb 23 '18 at 22:35
  • $\begingroup$ Yep, check the book "Differential Calculus (H. Cartan)", he explains that well for multilinear maps in general.. $\endgroup$ – PtF Feb 24 '18 at 23:32
  • $\begingroup$ For people reading this post, I'll let this link that further enlightens the discussion about bilinearity and Frechét. math.stackexchange.com/questions/65686/… $\endgroup$ – jpugliese Feb 25 '18 at 23:43
  • $\begingroup$ @PtF could I do a bound like this: $\|\phi(v_1, v_2)\|\leq C||v_1,v_2|| \leq C(||v_1|| + ||v_2||)$ and since $||v||$ $\rightarrow 0$, we have that $||v_1||$ and $||v_2||$ tends to zero, then we have that the proposed limit converges to zero? $\endgroup$ – jpugliese Feb 26 '18 at 22:33

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