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 $.

Thank you for your help.