Basic operations are continuous on Schwartz space

By [F1], $p_{\alpha \beta }({\mathrm{\partial }}^{\delta }f)=p_{\alpha ,\beta +\delta }(f)$. For a monomial multiplier $x^{\delta }$, its product-rule term indexed by $\gamma \le \beta ,\delta$ is $\left(\genfrac{}{}{0px}{}{\beta }{\gamma }\right)\delta !\mathrm{/}(\delta -\gamma )!$ times $x^{\delta -\gamma }{\mathrm{\partial }}^{\beta -\gamma }f$, giving the bound by the corresponding finite sum of $p_{\alpha +\delta -\gamma ,\beta -\gamma }(f)$. A polynomial is a finite sum of these monomials. For $T_{a}f(x)=f(x-a)$, put $y=x-a$ and expand $(y+a{)}^{\alpha }$; then $p_{\alpha \beta }(T_{a}f)\le {\sum }_{\gamma \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\gamma }\right)\mathrm{\mid }{a}^{\alpha -\gamma }\mathrm{\mid }{p}_{\gamma \beta }(f)$.

For $M_{b}f(x)=e^{2\pi ib\cdot x}f(x)$, [F1] and [F2] give $p_{\alpha \beta }(M_{b}f)\le {\sum }_{\gamma \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\gamma }\right)\mathrm{\mid }(2\pi ib{)}^{\gamma }\mathrm{\mid }{p}_{\alpha ,\beta -\gamma }(f)$. Reflection $Rf(x)=f(-x)$ has $p_{\alpha \beta }(Rf)=p_{\alpha \beta }(f)$, and conjugation has the same identity, since coordinate derivatives commute with real and imaginary parts. These formulas also prove the asserted linearity or antilinearity.

All bounds in steps 1.1 and 1.2 are finite seminorm sums; requiring their finitely many input seminorms to be sufficiently small proves continuity at zero directly from the topology, and linearity or antilinearity translates this to every point. Product Leibniz further gives $p_{\alpha \beta }(fg)\le {\sum }_{\gamma \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\gamma }\right)p_{\alpha \gamma }(f)p_{0,\beta -\gamma }(g)$, proving closure. At $(f_0,g_0)$ write $fg-f_0{g}_0=(f-f_0)g_0+f_0(g-g_0)+(f-f_0)(g-g_0)$ and apply this bound to all three terms. Each of the finitely many errors tends to zero with the relevant input seminorms, proving joint continuity and bilinearity.