How is the value of the norm of $f$ and $g$ calculated in this example?

Question

The example is given below:

How is the value of the norm of $f$ and $g$ calculated in this example? I tried the first one it gave me $\frac{\sqrt{5}}{\sqrt{3}}$ and the $g$ gave me $\frac{\sqrt{3}}{\sqrt{2}}.$

Answer 1

For the space of continuous functions over $(0,1)$, this inner product is defined analogously to the dot product when considering a vector space.

We know that the dot product of two vectors $\vec{u}$ and $\vec{v}$ is given as below:

$\vec{u} \cdot \vec{v} = ||\vec{u}|| ||\vec{v}|| \cos(t)$

It is immediately clear then that $\vec{u} \cdot \vec{u} = ||u||^2$

The inner product is a generalization of this, and that's how the norm is defined as $||h||= \sqrt{\int_{0}^{1} h(x)^2dx}$ in this case for any function $h$ in the space.