I am trying to understand the proof which gives weak closure of unit sphere is closed unit ball ($B$).
Let $x_0\in B$ and $G=\bigcap_{i=1}^n\{x : |\ell_i (x-x_0)<\delta_i\}$ is any open weak neighborhood of $x_0$ where $\ell_i$s are continuous functionals and $\delta_i$s are positive real numbers.
Proof says: then there exists $0\neq y\in \bigcap_{i=1}^n Ker(\ell_i) $. I don't understand this part. I am not familiar with the notion codimension.
Any explanation in elementary level will be appreciated.
Thanks a lot.
I suppose we work in a space $X$ with infinite dimension. This question is about pure linear algebra. Nowhere is topology relevant.
The claim is that $\bigcap_{i=1}^n \ker(\ell^i)\ne 0$. To see this, define the linear map $$T: X \to\mathbb{K}^n: x \mapsto (\ell^1(x), \dots, \ell^n(x)).$$
Then $\ker(T) = \bigcap_{i=1}^n \ker(\ell^i).$ By the isomorphism theorem of vector spaces, we get an injective linear map $$X/\ker(T) \to \mathbb{K}^n$$ and thus in particular $$\dim(X/\ker(T)) \le n.$$ If $\ker(T) = 0$, then the left hand side is $\dim(X) = \infty$, which is impossible. Hence, $\ker(T) \ne 0$. In fact, $\ker(T)$ has infinite dimension.