A Normed Linear Space on the Set of Continuous Functions on [0,1]

Consider the set of continuous function on

$[0,1]$
.
The set of functions is a vector space and we can define a norm on this space as
$||f(x)|| = \sqrt{\int^1_0 (f(x))^2 dx}$

We show that this satisfies all the properties of a norm.
1.
$||f(x)|| \geq 0$
and
$||f(x)|| =0 \leftrightarrow f(x)=0$
everywhere on
$[0,1]$
.
$||f(x)|| = \sqrt{\int^1_0 (f(x))^2 dx} \geq \sqrt{\int^1_0 0 dx} =0$

Suppose
$f(x) \neq 0$
for some interval
$[a,b] \subseteq [0,1]$
so that
$|f(x)| > \epsilon > 0$
for
$x \in [a,b]$

Then
$||f(x)|| > \sqrt{(b-a) \epsilon^2}= \epsilon \sqrt{b-a}$
so that if
$||f(x))| =0, \: f(x)=0$
2.
$||kf(x)|| =k ||f(x))||$
for any
$k, \: x$

$||k f(x))|| = \sqrt{\int^1_0 (kf(x))^2}= k \sqrt{\int^1_0 (f(x))^2 dx } = k ||f(x))||$
3.
$||f(x)+g(x)|| \leq ||f(x)|| + ||g(x)||$

\begin{aligned} ||f(x)+g(x)||^2 &= \int^1_0 (f(x)+g(x))^2 dx \\ &= \int^1_0 (f(x))^2 + 2 f(x) g(x) + (g(x))^2 dx \\ & \leq \int^1_0 (f(x))^2 dx + 2 \int^1_0 |f(x)| |g(x)| dx + \int^1_0 (g(x))^2 dx \\ & \leq \int^1_0 (f(x))^2 dx + 2 \sqrt{\int^1_0 (f(x))^2 dx} \sqrt{\int^1_0 (g(x))^2 dx} + \int^1_0 (g(x))^2 dx \\ &= (||f(x)||+||g(x||)^2 \end{aligned}

Square rooting both sides gives
$||f(x)+g(x)|| \leq ||f(x)|| + ||g(x)||$