Notion of weak convergence on a normed space without inner product

Question

It seems weak convergence, $x_n \rightharpoonup x$, means that $\displaystyle\lim_{n\to\infty} \langle x_n,x\rangle = \langle x,x\rangle$. Now if we fix the left position of the inner product, then we have the functional $f_a(x) = \langle x,a\rangle$. This reconstructs the inner product as a functional, and we get weak convergence to mean that $\lim_{n\to\infty} f_a(x_n)=f_a(x)$

First of all is that acceptable logic?

Secondly, is there a notion of weak convergence in a normed space that cannot have an inner product?

Answer 1

A normed space with an inner product that induces the norm is already a unitary/pre-Hilbert space.

And no, your notion of weak convergence is wrong: the correct version is $\lim_{n\to\infty}\langle x_n,y\rangle = \langle x,y\rangle$ for all $y\in X'$, where $X'$ is the dual space of $X$ (so in particular $X=X'$ if $X$ is a Hilbert space). This definition makes perfect sense even without any inner product structure.