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is it possible to define the derivative of a function of countable variables?

I found differential calculus of function with a finite number of variables, or differential calculus in Banach spaces (uncountable number of dimensions) . but what can be done in countable dimensions? functions of variables in $\mathbb{R}^\mathbb{N}$?

user88163
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    Depends on what you mean by dimension- infinite-dimensional Banach spaces do not have countable Hamel bases, although they can have countable Schauder bases. – AGM Oct 01 '13 at 22:35
  • @AnthonyCarapetis, maybe it is a silly question, but how do we know that there isn't any norm (or maybe there is no norm?) in $\mathbb{R}^{\mathbb{N}}$, such that $\mathbb{R}^{\mathbb{N}}$ is Banach? – Tomás Oct 01 '13 at 23:27
  • @Tomás: see e.g. http://math.stackexchange.com/a/217843/28513 – Anthony Carapetis Oct 02 '13 at 00:42
  • Very interesting, thank you @AnthonyCarapetis – Tomás Oct 02 '13 at 00:45
  • @Tomás: I actually think I've made a mistake - it is the countable direct sum of $\mathbb R$ that has no Banach structure. I think the countable product in fact has uncountable Hamel dimension. – Anthony Carapetis Oct 02 '13 at 00:46
  • @AnthonyCarapetis, is the direct sum the same as the space of all polynomious? – Tomás Oct 02 '13 at 01:09
  • @Tomás: yes, it's those elements of the product having only finitely many non-zero terms. I guess the full product $\mathbb{R^N}$ is the set of formal power series if you take this perspective. – Anthony Carapetis Oct 02 '13 at 01:11

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Suppose $f:X\to\mathbb{R}$ is a function and we want to define its derivative in a similar way as that we do when the function is defined in $\mathbb{R}^n$, i.e. we want to mimic the following limit (where $g:U\subset\mathbb{R}^n\to\mathbb{R}$) $$\lim_{t\to 0}\left|\frac{g(x+ty)-g(x)}{t}-g'(x)y\right|=0 \tag{1}$$

From $(1)$ we see that we need two things: the first one is a way to measure the length of the expression $\left|\frac{g(x+ty)-g(x)}{t}-g'(x)y\right|$. The other one, is the fact that $g(x+y)$ must be well defined. We can solve these two issues by asking that $X$ is a normed vector space.

Now, we can say that a function has Gâteaux derivative in the point $x$ with direction $y$ if and only if there exist a bounded linear functional $f'(x):X\to\mathbb{R}$ such that $$\lim_{t\to 0}\left\|\frac{f(x+ty)-f(x)}{t}-f'(x)y\right\|=0$$

where $\|\cdot\|$ is the norm in $X$. If $f$ is Gâteaux differentiable in some point $x$ and the Gateaux derivative is continuous in this point $x$, then we say that $f$ is Fréchet differentiable in $x$.

Now, there is plenty of examples of normed spaces $X$ which are infinite dimensional and you can derivate, for example, you can take the set $X=C([0,1])=\{u:[0,1]\to\mathbb{R}:\ \mbox{$u$ continuous}\}$. Define $f:X\to X$ by $$f(u)=u^2$$

You can verify that $f'(u)v=2uv$, $u,v\in X$.

Remark: Note that the space $\ell^p=\{x=(x_1,...,x_n,...):\ \sum_{i=1}^\infty |x_i|^p<\infty\}$ for $p\in [1,\infty)$ is included here.

Tomás
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