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