norm space(SEQUENCE AND SERIES OF FUNCTIONS)

 After studying some global properties of series of functions, the concept of norm space furnished a new version to describe them.

7.14 Definition If $X$ is a metric space, $\mathscr{C}(X)$ will denote the set of all complex-valued, continuous, bounded functions with domain $X$.

  We associate with each $f\in\mathscr{C}(X)$ its supremum norm

$$ \Vert f\Vert=\sup\limits_{x\in X}\vert f(x)\vert. $$

If $h=f+g$, then

$$ \vert h(x)\vert \leq \vert f(x)\vert+\vert g(x)\vert\leq \vert f\vert+\vert g\vert $$

for all $x\in X$. Hence

$$ \Vert f+g\Vert \leq \Vert f\Vert+\Vert g\Vert. $$

  If we define the distance between $f\in\mathscr{C}(X)$ and $g\in\mathscr{C}(X)$ to be $\Vert f-g\Vert$, it follows that $\mathscr{C}(X)$ becomes a metric space.

  A new version mentioned above, is that we can restate some previous theorems as follows:

  A sequence $\{f_n\}$ converges to f with respect to the metric of $\mathscr{C}(X)$ if and only if $f_n\rightarrow f$ uniformly on $X$.

  This is an example of the supremum norm. We could also define some different norms, like:

$$ \Vert f\Vert=(\int_{0}^{1}\vert f(x)\vert^{p}dx)^{1/p}, $$

where $X=[0,1]$.