*Definition [#je6b6573]
-A line-bounded convex subset K of a linear topoIogical space X is said to be σ-line-compact if the following holds: let &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 , &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/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 [#s61a5cb5]
-Every σ-line-compact set is [[line-compact]].

*Reference [#jf0b24b7]
-Rosenthal, Haskell, ''L^1-convexity. Functional analysis'', 156--174, Lecture Notes in Math., 1332, Springer, Berlin, 1988.