Proof \( C \left [ a, b \right ] \) with the Norm \( \left | \left | \cdot \right | \right | _{\infty} \) is a Normed Space

Clearly the set of continuous function \( C \left [ a, b \right ] \) is a vector space. Now, we have to prove that the function \( \left | \left | \cdot \right | \right | _{\infty} :  C \left [ a, b \right ] \to \mathbb{R}  \) such that 

\( \left | \left | f \right | \right | _{\infty} = \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| \) for every function \( f \in C \left [ a, b \right ] \)

is really a norm.

The first property we have to prove is:

\( \left | \left | f \right | \right | _{\infty} = 0 \) if and only if \( f = 0 \) is the zero function. 

Obviously, if \( f \) is the zero function, 

\( \left | \left | f \right | \right | _{\infty} =  \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| =  \displaystyle \max_{a\leq x\leq b} \left| 0 \right| = 0 \).

Now, assume  \( f \) is not a zero function, therefore there exists \( c \in \left [ a, b \right ] \) such that \( f(c) \neq 0 \). Next, we obtain

\( \left | \left | f \right | \right | _{\infty} =  \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| \geq \left| f(c) \right| > 0 \)

\( \Rightarrow \left | \left | f \right | \right | _{\infty} \neq 0 \).

Thus, we conclude that \( \left | \left | f \right | \right | _{\infty} = 0 \) if and only if \( f \) is the zero function. 

Next, we have to prove for every \( f \in   C \left [ a, b \right ] \) and \( k \in \mathbb{R} \),

\( \left | \left | kf \right | \right | _{\infty} = \left | k \right | \left | \left | f \right | \right | _{\infty}  \).

Take any \( f \in   C \left [ a, b \right ] \) and \( k \in \mathbb{R} \). Since \( f \) is a continous function then \( \left | f  \right | \) is also continous on \( \left [ a, b \right ] \). Now, since \( \left | f  \right | \) is continous on the compact set \( \left [ a, b \right ] \), \( \left | f  \right | \) has maximum (this is also telling us that  \( \left | \left | \cdot \right | \right | _{\infty} \) is well defined). Let \( c \in \left [ a, b \right ] \) such that \( \left | f(c)  \right | \) is the maximum value.

Given \( x \in \left [ a, b \right ] \), we obtain

\( \left | kf(x) \right | = \left | k \right | \left | f(x) \right | \leq  \left | k \right | \left | f(c) \right | \),

and 

\( \left | kf(c) \right | = \left | k \right | \left | f(c) \right | \).

It shows that \( \left | k \right | \left | f(c) \right | \) is the maximum value of \( \left | kf \right | \). In other words

\( \left | \left | kf \right | \right | _{\infty} =  \displaystyle \max_{a\leq x\leq b} \left| kf(x) \right|  =  \left | k \right | \left | f(c) \right | =   \left | k \right |\left | \left | f \right | \right | _{\infty} \).

Hence, the second property has been proved.

Lastly, we have to prove the triangle inequality. It is easy to see that for any given \( f,g \in   C \left [ a, b \right ] \), we obtain for every \( x \in \left [ a, b \right ] \),

\( \left | f(x) + g(x) \right | \leq \left | f(x) \right | + \left | g(x) \right | \leq  \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| +  \displaystyle \max_{a\leq x\leq b} \left| g(x) \right| =  \left | \left | f \right | \right | _{\infty} +  \left | \left | g \right | \right | _{\infty}\).

Therefore,  \( \left | \left | f \right | \right | _{\infty} +  \left | \left | g \right | \right | _{\infty} \) is an upper bound of \( \left | f + g \right | \). Thus, we have proved the triangle inequality

\( \left | \left | f + g \right | \right | _{\infty} = \displaystyle \max_{a\leq x\leq b} \left| f(x) + g(x) \right| \leq \left | \left | f \right | \right | _{\infty} +  \left | \left | g \right | \right | _{\infty} \).

We have proved all of the properties that necessary for \( \left | \left | \cdot \right | \right | _{\infty} \) to be a norm on the vector space \( C \left [ a, b \right ] \). So, \( C \left [ a, b \right ] \) with the norm \( \left | \left | \cdot \right | \right | _{\infty} \) is a normed space. \( \blacksquare \)