1. Weak Convergence
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In this chapter we first recall without proofs the basic facts from functional
analysis concerning weak convergence and weak compactness for functions
and measures. We next systematically sharpen certain real analysis tools, to
be employed later on.
A. Review of Basic Theory
Here and hereafter we will denote by U an open, bounded, smooth subset
of R (n 2). Assume 1 q oo, q = q/q - 1.
DEFINITION.
Á sequence {fk)k_ c L (U) converges weakly to / e
Lq{U),
written
fk-*f in
L9(U),
provided
(1.1) / fkgdx-+ \ fgdx asA:
Ju Ju
for each g e L (U).
THEOREM
1
(BOUNDEDNESS OF WEAKLY CONVERGENT SEQUENCES).
Assume
fk- f in
Lq
(U). Then
(i) {fk}T is bounded in L (U)
and
(ii) 11/1 1
q
liminf||/J|
q
.
é
q q
In view of (i) we see that if fk -* f in L (U) and gk g in L (U),
then
/ fkgkdx-+ É fgdx.
Ju Ju
Á refinement of (ii) holds: if 1 q oo, fk-*finL (U) and
11/11 q = lim ||/. ||
q
then
fk - /strongly in L {U).
http://dx.doi.org/10.1090/cbms/074/02
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