1. Weak Convergence 0 0 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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