Schwartz topology and convergence

For the seminorms of Schwartz space and its seminorms, a basic neighbourhood of $f_0\in \mathcal{S}$ is $\{f:p_{{\alpha }_r{\beta }_r}(f-f_0)<{\epsilon }_r(1\le r\le m)\},m<\mathrm{\infty },{\epsilon }_r>0.$ The empty intersection is the whole space. The topology consists of unions of these neighbourhoods. A sequence $f_k$ converges to $f$ precisely when $p_{\alpha \beta }(f_k-f)\to 0$ for every pair of multi-indices: necessity follows from the one-condition neighbourhoods, and sufficiency follows by taking the maximum of the finitely many convergence thresholds in a basic neighbourhood.

A complex topological vector space is called locally convex here when zero has a base of convex balanced sets; balanced means $\mathrm{\mid }c\mathrm{\mid }\le 1$ implies $cV\subseteq V$. The displayed sets about zero are convex and balanced by the seminorm inequalities. They also show addition is continuous, by halving each tolerance. Scalar multiplication is jointly continuous: near $(c_0,f_0)$, restrict $\mathrm{\mid }c-c_0\mathrm{\mid }<1$ and use $p_{\alpha \beta }(cf-c_0{f}_0)\le (\mathrm{\mid }{c}_0\mathrm{\mid }+1)p_{\alpha \beta }(f-f_0)+\mathrm{\mid }c-c_0\mathrm{\mid }{p}_{\alpha \beta }(f_0)\mathrm{.}$ There are finitely many bounds to enforce. Finally $p_{00}(f-g)>0$ for distinct functions; balls of radius less than half this number separate them. Thus this is a Hausdorff locally convex topological vector space. All these verifications use only finite choices.