The uniform boundedness principles for γ-max-pseudo-norm-subadditive and quasi-homogeneous operators in F ∗ spaces

In this paper, we prove that every F ∗ space (i.e., Hausdorff topological vector space satisfying the first countable axiom) can be characterized by means of its “standard generating family of pseudo-norms”. By using the standard generating family of pseudo-norms P, the concepts of P-bounded set and γ-maxpseudo-norm-subadditive operator in F ∗ space are introduced. The uniform boundedness principles for family of γ-max-pseudo-norm-subadditive and quasi-homogeneous operators in F ∗ spaces are established. As applications, we obtain the corresponding uniform boundedness principles in classical normed spaces and Menger probabilistic normed spaces. c ⃝2015 All rights reserved.