Proposition:Let X be a given vector space and Γ be the set of linear functional on X.Let X endowed with a weak topology with respect to Γ.Then X is a topological vector space.
What I need to show is (x,y)∈X×X→x+y is continuous and (α,x)∈C×X→αx is continuous.However,generally a weak topology assure only the continuity of each linear functions on X.So I don’t know how to show the proposition.Please give me a hint or answer.
Well, that the topology is generated by the continuous functionals means that for a given net $(x_{\alpha})_{\alpha\in A},$ $x_{\alpha}\to x$ if and only if $f(x_{\alpha})\to f(x)$ in $\mathbb{C}$ for every $f\in \Gamma$.
So assume $x_{\alpha}\to x,$ $y_{\alpha}\to y$ and $\lambda_{\alpha}\to \lambda$ (the latter limit taken in $\mathbb{C}$). Then, for any $ f\in \Gamma$, we have $$ f(x_{\alpha}+y_{\alpha})=f(x_{\alpha})+f(y_{\alpha})\to f(x)+f(y) \\ f(\lambda_{\alpha}x_{\alpha})=\lambda_{\alpha}f(x_{\alpha})\to \lambda f(x), $$ where we've used that the ring operations on $\mathbb{C}$ are continuous. This establishes that yes, the weak topology makes $X$ a topological vector space.