In this paper,We show that 1~∞(X)-evaluation uniform convergence of operator series can be described completely by the essential bounded subset of 1~∞(X),this conclusion improves the J.

we obtained the following results:1 With the concept of uniformly forward subsets of the spaces c(X),essentially bounded subsets of the spaces l∞(X) and uniformly exhaustive subsets of the spaces l~p(X) and the description of operator series family of l_p~β(X),we get a equipollence proposition about uniformly forward,essentially bounded and uniformly exhaustive subsets.

The nonemptyness and boundedness of the solution set for the second-order cone complementarity problem with a Cartesian P_*(κ) mapping are investigated.

In general,the convergent sequence and bounded set are concepts only in topological spaces.

Moreover,for the space X satisfying a condition (Q),each bounded set in C 0(X) admits a center if X is quasi-uniformly convexity.

Introducing the difinition of bounded set in fuzzy paranormed linear space , this paper discusses several concerned characteristics and studies the local boundedness of fuzzy paranormed linear space.

We produce a family of bounded sets Δ u .

In quasi-normed space, the limit of normed γ-quasi-subadditive operator sequence or normed γ-max-quasi-subadditive operator sequence being equi-continunous operator sequence in quasi-normed space is bounded in any quasi-bounded set, and its normed γ-quasi-subadditivity or normed γ-max-quasi-subadditivity is invariable.