Denseness for norm attaining operator-valued functions
Abstract In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss' famous result on the denseness of norm attaining operators. Specifically, we show given any A∈ L ( H ) there is a sequence of rank-1 operators K n such that A+K n is norm attaining for each n and K n converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operator-valued functions.
A proof that every Banach space is subreflexive
A real or complex normed space is subreflexive if those f unctionals which attain their supremum on the unit sphere S of E are normdense in E*, i.e., if for each ƒ in £ * and each e > 0 there exist g in E* and x in S such that \g(x)\ =\\g\\ and ||/—g|| <€. There exist incomplete normed spaces which are not subreflexive [ l ] 1 as well as incomplete spaces which are subreflexive (e.g., a dense subspace of a Hubert space). I t is evident that every reflexive Banach space is subreflexive. The theorem mentioned in the title will be proved for real Banach spaces; the result for complex spaces follows from this by considering the spaces over the real field and using the known isometry between complex functionals and the real functionals defined by their real parts. We first cite a lemma which states, roughly, that if the hyperplanes determined by two functionals ƒ and g (of norm one) are nearly parallel, then one of ||/—g||, | | /+g | | must be small.
PHOTOGRAMMETRY AND ITS USES IN HIGHWAY PLANNING AND DESIGN
PHOTOGRAMMETRY MIGHT BE DESCRIBED AS THE PROCESS OF CONVERTING PHOTOGRAPHS INTO CONTOURED TOPOGRAPHIC MAPS. FORMERLY GROUND SURVEY METHODS WERE CONSIDERED THE ONLY RELIABLE MEANS OF OBTAINING ACCURATE TOPOGRAPHIC INFORMATION. IT IS NOW POSSIBLE TO DEVELOP TOPOGRAPHIC MAPS OF SPECIFIED ACCURACY BY MEANS OF AERIAL PHOTOGRAPHS WHICH ARE THEN SUBJECTED TO A SERIES OF PHOTOGRAMMETRIC PROCESSES SO THAT MAPS MAY BE DRAWN AT SCALES USEFUL IN HIGHWAY LOCATION AND DESIGN. /AUTHOR/