Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ continuous with respect to $\sigma(X,P)$.
My question is this. Is $\sigma(X,P)$ compatible with the algebraic structure of $X$? If yes, can you please explain me why.
Convergence in the weak topology is given by convergence in all of the seminorms in $P$.
Suppose that $x_j\to x$, $y_j\to y$ weakly. For any $p\in P$, $$ p(x_j+y_j-(x+y))=p((x_j-x)+(y_j-y))\leq p(x_j-x)+p(y_j-y)\to0. $$ So addition is continuous. If $\lambda_j\to\lambda$ in $\mathbb C$, we have $$ p(\lambda_jx_j-\lambda x)\leq |\lambda_j|\,p(x_j-x)+|\lambda_j-\lambda|\,p(x)\to0. $$ This shows that scalar multiplication is continuous.