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σ-line-compact

Last-modified: 2010-08-31 (火) 09:42:50

Definition Edit

  • A line-bounded convex subset K of a linear topoIogical space X is said to be σ-line-compact if the following holds: let imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%28u_n%29%5c%20%5cmbox%7band%20%7d%28v_n%29%20%5c%5d%7d%25.png be sequences in K so that there exist imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20k_1%2ck_2%2c%5cldots%5c%20in%5c%20K%2c%5c%20%5clambda_1%2c%5clambda_2%2c%5cldots%20%5c%5d%7d%25.png positive scalars with imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5csum%5cmbox%7b%7d%5e%7b%5cinfty%7d_%7bj=1%7d%5clambda_j=1%2c%200%3c%5ctheta%3c1%2c%5c%20%5cmbox%7band%20%7dx%5cin%20K%20%5c%5d%7d%25.png so that for all n , imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20u_n=%5cgamma_n%5e%7b-1%7d%5csum%5cmbox%7b%7d%5e%7bn%7d_%7bj=1%7d%5clambda_j%20k_j%20%5c%5d%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x=%5ctheta%5cgamma_n%20u_n%20+%281-%5ctheta%5cgamma_n%29v_n%20%5c%5d%7d%25.png where imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cgamma_n=%5csum%5cmbox%7b%7d%5en_%7bj=1%7d%5clambda_j%20%5c%5d%7d%25.png Then both imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%28u_n%29%5c%20%5cmbox%7band%20%7d%28v_n%29%20%5c%5d%7d%25.png converge to elements of K.

Property Edit

Reference Edit

  • Rosenthal, Haskell, L^1-convexity. Functional analysis, 156--174, Lecture Notes in Math., 1332, Springer, Berlin, 1988.