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Unformatted text preview: sing its completeness). However u = u� as before, so u → C0 (Rn ).
� In particular we see that
(10.6) H � (Rn ) = � H m (Rn ) � C � (Rn ) . m These functions are not in general Schwartz test functions.
Proposition 10.4. Schwartz space can be written in terms of weighted
Sobolev spaces
�
(10.7) S (Rn ) = ⊂x≤−k H k (Rn ) .
k LECTURE NOTES FOR 18.155, FALL 2004 65 Proof. This follows directly from (10.5) since the left side is contained
in
�
k
⊂x≤−k C0 −n (Rn ) � S (Rn ).
k � Theorem 10.5 (Schwartz representation). Any tempered distribution
can be written in the form of a ﬁnite sum
�
�
0
(10.8)
u = x� Dx u�� , u�� → C0 (Rn ).
��m
� �m or in the form
(10.9) u= � �
0
Dx (x� v�� ), v�� → C0 (Rn ). ��m � �m Thus every tempered distribution is a ﬁnite sum of derivatives of
continuous functions of p oynomial growth.
Proof. Essentially by deﬁnition any u → S � (Rn ) is continuous with re
spect to one of the norms ∈⊂x≤k α∈C k . From the...
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This note was uploaded on 01/22/2014 for the course MATH 18.155 taught by Professor Melrose during the Fall '13 term at MIT.
 Fall '13
 Melrose
 Derivative

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