A Question Regarding a Norm over Tempered Distributions

Question

I am having trouble understanding a definition regarding tempered distributions from Bahouri's "Fourier Analysis and Nonlinear PDEs". In definiton 1.26, the author defines the subspace $\mathcal{S}^{'}_h$ of tempered distributions through a condition related to an $L^\infty$ norm of a convolution of a Schwartz function with a tempered distributon, which is again a tempered distribution. Yet, it is not clear to me how we can consider a $L^\infty$ norm over a tempered distribution. How does one makes sense of this ?

Answer 1

The shortest might be to proceed by duality with $L^1$ $$ \|u\|_{L^\infty} = \sup_{\varphi\in\mathcal S, \|\varphi\|_{L^1}\leq 1} \langle u,\varphi \rangle. $$ Of course, when the above quantity is finite, then $u$ will be identifiable with a $L^\infty$ function anyway.