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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.  
\[||f(x)+g(x)|| \leq ||f(x)|| + ||g(x)||\]

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

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