I'm trying to give an example to show the following:
Not every bounded linear functional on $L^3([0,1],\mu)$ is the restriction of a bounded linear functional on $L^2([0,1],\mu)$. (Here $\mu$ is Lebesgue measure)
So far for finite measure spaces with $0<p<q\leq\infty$ I have that $L^q\subset L^p$ (from Folland) so the restriction makes sense. Restricting a bounded linear functional from $L^p$ to $L^q$ shouldn't be a problem, it retains boundedness and linearity from the fact that $L^q\subset L^p$.
I found the Hahn-Banach theorem, but this seems to only give conditions for when one can extend a bounded linear functional. (I didn't find anything on if the conditions it gives are also necessary, in addition to being sufficient)
I'd appreciate an example and/or good places to look on general theory. (Perhaps I just didn't google the right terms)