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I'm pretty confused with the idea of local maximum in function spaces. Normally having a null Fréchet derivative is a necessary but not sufficient condition for being a local maximum.

Computing the derivative

let $ f:\mathbb{R} \mapsto \mathbb{R}$ be a continuous function. And lets denote the space of such functions $C_{\mathbb{R,R}}$.

$$ F: C_{\mathbb{R,R}} \mapsto C_{\mathbb{R,R}} $$ $$ F: f \mapsto \sin(f)$$

let us compute its derivative at point f:

$$D_F(f)h = \lim_{t\to 0} {F(f+th) - F(f) \over t} $$

where:

  1. $ f \in C_{\mathbb{R,R}} $
  2. $ g \in C_{\mathbb{R,R}} $
  3. $ t \in \mathbb{R}$

then:

$$D_F(f)h = \lim_{t\to 0} {\sin(f+th) - \sin(f) \over t} $$ $$D_F(f)h = \lim_{t\to 0} {\sin(f)\cos(th)+\cos(f)\sin(th) - \sin(f) \over t} $$ $$D_F(f)h = \lim_{t\to 0} {-h\sin(f) (1 - \cos(th))+h\cos(f)\sin(th) \over th} $$

So using: $$\lim_{x\to 0} {\sin(x)\over x} = 1 $$ $$\lim_{x\to 0} {1-\cos(x)\over x} = 0 $$

it reduces to:

$$D_F(f)h = h\cos(f)$$

Local maximum

we obviously have:

$$ 0 \leq ||F(f)||_\infty \leq 1$$

Hence:

if $||F(f)||_\infty = 1$, then f is a local maximum.

So any function $f$ such that $f(x) \equiv \pi \pmod \pi$ has a solution is a local maximum.

Null Fréchet derivative

$$D_F(f) = \cos(f) = 0 \Rightarrow \exists k \in \mathbb{N}, \forall x \in \mathbb{R}, f(x) = k\pi $$

such constant functions are indeed local maximums, but not the only ones. So instead of getting a superset containing all my local maximums, I get a strict subset of it from the nullity of my Fréchet derivative.

Question

As I'm pretty sure the mathematics I'm taught are right and I'm wrong... Where am I wrong ?

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  • $\begingroup$ What is the definition of the maximum of a mapping from $C(\mathbb R)$ to $C(\mathbb R)$? You seem to maximize $\|F(f)\|_\infty$ - the $\infty$-norm is not that differentiable. And $\cos(f)=0$ implies $\forall x\in \mathbb R$ $\exists k\in N$ such that $f(x)=k\pi$. Now find all such continuous functions... $\endgroup$ – daw May 15 '14 at 13:10
  • $\begingroup$ @daw "the ∞-norm is not that differentiable", you mean there is a condition about the norm of my banach space for the fréchêt derivative to exists ? Or for a maximum do be defined ? $\endgroup$ – user2346536 May 15 '14 at 13:22
  • $\begingroup$ @daw "Now find all such continuous functions" $ \forall x f(x) = kπ $ seems a good shot to me. Constant functions basically. $\endgroup$ – user2346536 May 15 '14 at 13:32
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    $\begingroup$ You seem to maximize $\phi(f):=\|F(f)\|_\infty$, but you only compute the derivative of $F$, but you do not check differentiability of $\phi$. $\endgroup$ – daw May 15 '14 at 13:38
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    $\begingroup$ @daw You mean that, finding a maximum of a function $ F:C_{\mathbb{R,R}} \mapsto C_{\mathbb{R,R}} $ does not really makes sens as $C_{\mathbb{R,R}}$ is not ordered. And so when I choosed my norm $||||_\infty$, I in fact was trying to maximize $||F(f)||_\infty$ ? Which is $ \phi:C_{\mathbb{R,R}} \mapsto \mathbb{R} $ where R is order and $\phi(f) \leq \phi(f) $ does make sens. $\endgroup$ – user2346536 May 15 '14 at 13:44
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The problem is to maximize $$ \|F(f) \|_\infty. $$ However, the maximum is not differentiable. In order to perform the analysis, the function $f\mapsto\|F(f) \|_\infty$ needs to be differentiable

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