Convex function space How do you show a function space is convex? I understand this notion in the concept of norms, where in the plane I've sketched the $L_1,L_2,L_\infty$ norms, where clearly $L_1,L_\infty$ are not strictly convex and $L_2$ is. Intuitively it would make sense that the $L_1,L_\infty$ function spaces are not strictly convex (similar to their norm) and the $L_2$ function space is strictly convex, but how can you formally show this? (seems different than showing their norms are strictly convex, which I've seen the forums for.)
 A: You are confusing few things. First of all, you said you've sketched $L_p$ norms in plane - which is unlikely. You probably meant $l_p$ norms being restricted to the $\mathbb{R}^2$. To clarify what I mean:
$$
||f||_{L^p(\Omega, \mu)} := \bigg( \int\limits_{\Omega} |f|^p d\mu \bigg )^{\frac{1}{p}}
$$
Here $\Omega$ is an open subset of $\mathbb{R}^n$, and $\mu$ is a measure. As you can see, it's not something to be sketched, and definitely not on the plane. What you meant is probably:
$$
||(x_1,x_2)||_{l^2} := \sqrt{x_1^2+x_2^2}
$$
This is a special, finite-dimensional case of general $l^p$ norms, which are defined as:
$$
||x||_{l^p}=\bigg( \sum_{i=1}^{\infty} |x_i|^p \bigg )^{\frac{1}{p}}
$$
Here $x=(x_1,x_2,...)$. Once again, it's not worth trying to visualize it, even though it coincides with Euclidean norm in finite dimensional cases.
About convexity part: being convex is not a property a space can hold on its own. It's rather that some subset or subspace $X$ of space $Y$ may be convex in $Y$. For example, an unit disc is convex in $\mathbb{R}$ w.r.t. the Euclidean norm you had in mind. So your question would rather be How can functional subspaces be convex in functional spaces?. Then again, it's the same definition. One can consider, for example, consider unit ball in $L^1(\mathbb{R}^n)$, that is the set $\{f \in L^1(\mathbb{R}^n) \ s.t. ||f||_{L^1} \le 1\}$. Then to test if it's compact is to prove that every convex combinations of two such functions also has norm less or equal to 1.
The broader context in which convexity and functional spaces come up together is a concept of locally convex topological vector spaces, which is a generalization of normed spaces. Those are usually defined with topology generated by family of semi-norms. There's also a definition making use of convex sets, which you can check on Wikipedia - it's actually written quite well.
