Suppose $E$ is a Banach space, and $X$ is a random vector in $E$. Now, using $X$ we can define a symmetric bilinear form on $E^*$, associated with it: $\langle f, g\rangle_X = E(f(X)g(X))$. $E^*$ is here the space of all bounded functionals on $E$. Are the following statements equivalent:
1)For all closed proper subspaces $E_1 < E$ $P(X \in E_1) < 1$;
2)$E^*$ is a Hilbert space with scalar product $\langle . , . \rangle_X$?
If not - which one of them is stronger?
Note, that for «one-dimensional» Banach space $\mathbb{R}$ they are indeed equivalent.
They are equivalent. It is easy to see that $ \langle f, g \rangle_X$ is an inner product iff $P(X \in E_1)<1$ whenever $E_1$ is the kernel of some $f \in E^{*}\setminus \{0\}$. Using Hahn Banach Theorem we can show that any proper closed subspace is contained in the kernel of some non-zero $f$ and this proves the equivalence.