# real analysis – Equivalence of norms on the Schwarz space

Consider the following standards on the Schwarz space, for $$1 leq q leq infty$$
$$GreenEnglish _ { alpha, beta, p} = lGreen x ^ { alpha} partial ^ beta f lVert_ {L ^ p}$$

I want to show that standards $$lVert cdot rGreen _ { alpha, beta, p}$$ define the same topology as the norm on the Schwarz space where the topology is given by $$lVert Green {alpha, beta} = lGreen (1+ | x |) ^ { alpha} partial ^ beta f lGreen_ {u}$$

Just show that one or the other of the seminar games provides a local base to 0 in the topology generated by the other.

For one direction we have fixed $$alpha, beta, epsilon$$and want to show that there are $$alpha, beta, epsilon$$ such as $${ Green cdot rGreen _ { alpha, beta, p} < epsilon } subset$$ $${ Green green cdot _ { alpha, beta} < epsilon }$$.

For this we truncate the integral $$int | f (x) | dx$$ by the unit disk and its complement, then $${ Green c Green {0,0, p} < epsilon }$$ is in $${ Green cdot rVert_ {0,0} < epsilon # bigcap { Green cdot rGreen_ {N, 0} < epsilon & # 39; & # 39; }$$ for big enough $$N$$. Since $$x ^ { alpha} partial ^ beta f$$ is in the Schwarz space though $$f$$ is in the Schwarz space we have $${ Green cdot rGreen _ { alpha, beta, p} < epsilon } subset$$ $${ Green green cdot _ { alpha, beta} < epsilon }$$.

I do not know how to deal with the other direction. It seems that I need to tie the standard of a Schwarz function $$f$$ by $$L ^ p$$ standard of some $$x ^ alpha f$$.