We extend the class of Banach sequence spaces constructed by Ledari, as presented in ''A class of hereditarily $ell_1$ Banach spaces without Schur property'' and obtain a new class of hereditarily $ell_p(c_0)$ Banach spaces for $1leq p<infty$. Some other properties of this spaces are studied.

1. Basic definitions 2. Riesz’ Lemma 3. Counter-examples for unique norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Baire’s theorem 7. Banach-Steinhaus/uniform-boundedness theorem 8. Open mapping theorem 9. Closed graph theorem 10. Hahn-Banach theorem Many natural spaces of functions, such as C(K) for K compact, and C[a, b], have natural str...

1. Introduction and notation. In this paper the word " local " is used in at least three different meanings. Our aim is to study local Banach spaces of Fréchet or other locally convex spaces, and it turns out that it is convenient to use the local theory of Banach spaces for this purpose. Recall that given a locally convex space E and a continuous seminorm p on E the completion of the normed sp...

The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...

1. Non-Banach limits C(R), C∞(R) of Banach spaces C[a, b] 2. Banach completion C o (R) of C c (R) 3. Rapid-decay functions, Schwartz functions 4. Non-Fréchet colimit C∞ c (R) of Fréchet spaces 5. LF-spaces of moderate-growth functions 6. Strong operator topology 7. Generalized functions (distributions) on R 8. Tempered distributions and Fourier transforms on R 9. Test functions and Paley-Wiener...

The main aim of this article is to introduce some difference sequence spaces with elements in a finite dimensional 2-normed space and extend the notion of 2-norm and derived norm to thus constructed spaces. We investigate the spaces under the action of different difference operators and show that these spaces become 2-Banach spaces when the base space is a 2-Banach space. We also prove that con...

2. Summary of the thesis 9 2.1. Di erentiability of convex functions and the respective classes of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2. Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3. Decompositions of nonseparable Banach spaces . . . . . . . . . . . . . . . 23 2.4. Summary of ...

Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space X means that, for every element u in the ...

Table of contents: Section 1: Overview, background 1 Section 2: Definition of Banach spaces 2 Section 3: Examples of Banach spaces 2 Section 4: L spaces 6 Section 5: L spaces 9 Section 6: Some spaces that are almost Banach, but aren’t 10 Section 7: Normed vector spaces are metric spaces 11 Section 8: How to make new spaces out of existent ones (or not) 13 Section 9: Finite-dimensional vector sp...