I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ satisfy that $\Vert f\Vert_\infty\le\sqrt{n}$. I think I'll have to find an example such that the above is true for all $n$.
I don't know where to start. I'm not sure if this helps, but in previous questions in the homework I showed that if $S$ is a subspace of $C[0,1]$ (which is closed at subspace of $L^2[0,1]$) then $\Vert f\Vert_{\infty}\le M\Vert f\Vert_2$ for all $f\in S$.
In the second part of this question, I will need to show that if $W$ is a $n$-dimensional subspace of $L^2[0,1]$, and all elements of $W$ are continuous functions, then there exists $f\in W$ s.t. $\Vert f\Vert_2=1$ and $\Vert f\Vert_\infty\ge\sqrt{n}$.