TITLE: What is the difference between $L^2$ norm and $\ell^2$ norm?
QUESTION [5 upvotes]: I can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?

REPLY [4 votes]: Regarding the switching I would like to add that $L^2([0,1])$ and $\ell^2$ are isomorphic as Hilbert spaces (as they are both separable and infinite-dimensional). That means: If $(f_n)_{n\in\mathbb N}$ is an orthonormal basis of $L^2([0,1])$, for example the basis $\{\exp(2\pi i n \,\cdot\,) : n \in \mathbb Z\}$, then 
\begin{align*}
   L^2([0,1]) &\to \ell^2\\
          f   &\mapsto (\left<f, e_n\right>)_n
\end{align*}
is an isometric isomorphism of Hilbert spaces, so especially $\|f\|_{L^2} = \|(\left<f, e_n\right>)_n\|_{\ell^2}$.