I have a set of elements in $L^2$ and I am looking for the norm and metric.
By reading Komogorov and Fomin, I deduce that the right norm and metric is:
Let $y_k \in Y=L^2$ defined on some domain $\mathscr{D}=[a,b]$, with variable $t$. We can then define the $2$-norm of $y_k$ by \begin{equation} \|y_k\|_2 := \sqrt{\int_a^b|y(t)|^2dt}| \end{equation}
Metric:
The norm induces the metric, which yields the distance between two elements $y_1,y_2\in Y$ (see theorem 5), which is given by
\begin{equation} \|y_2-y_1\|_2 := \sqrt{\int_a^b|y_2(t)|^2+|y_1(t)|^2dt}| \end{equation}
which is a linear functional giving the operation:
\begin{equation} G:Y\longrightarrow \mathbb{R} \end{equation}
which satisfies the following criteria....
However, I am not sure I am on the right track!
Can someone comment?
Thanks
The $L^2$ norm is, strictly speaking, applied to an equivalence class of functions, but to actually calculate the norm you use a representative (any representative) of the equivalence class.
Working with the space $L^2(X, \mathcal A, \mu)$, where $(X, \mathcal A, \mu)$ is any measure space, if $\tilde f \in L^2$ is an equivalence class of functions, and $f \in \tilde f$, then the $L^2$ space is constructed so that
\begin{align*} \Vert \tilde f \Vert_2 = \Vert f \Vert_2 = \left( \int \vert f \vert^2 \, d\mu\right)^{1/2} \end{align*}
(You can generalize this to $L^p$ space, where $p$ is a number other than 2.)
If you have any norm $\Vert \cdot \Vert$ on a normed vector space $E$, then as you write, that norm induces a metric, meaning that there is a certain thing you can do with that norm that results in a metric. Specifically, if $x, y \in E$, then $\Vert x - y \Vert$ is the induced metric. This is a function that computes distances; for more information you can read here, and the whole page on metric spaces more generally.
In the case of $L^2$, if $\tilde f, \tilde g \in L^2$ and $f \in \tilde f$ and $g \in \tilde g$, then
\begin{align*} \Vert \tilde f - \tilde g \Vert_2 = \left( \int \vert f - g \vert^2 \, d\mu\right)^{1/2} \end{align*}