We verify that the $p$-integrable Teichmüller space $T_{p}$ admits canonical complex Banach manifold structure for any $p \geq 1$. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of in terms $p$-Besov $p>1$.

Every Banach space X with the Banach-Saks property is reflexive, but the converse is not true (see [4, 5]). Kakutani [6] proved that any uniformly convex Banach space X has the Banach-Saks property. Moreover, he also proved that if X is a reflexive Banach space and θ ∈ (0, 2) such that for every sequence (xn) in S(X) weakly convergent to zero, there exist n1, n2 ∈ N satisfying the Banach-Saks p...

The real geometric properties of spaces of polynomials are discussed in [1, 6]. In particular, it is shown that the symmetric injective tensor product space ⊗̂n,s,εE is not strictly convex if E is a Banach space of dimE ≥ 2 and if n ≥ 2 holds. Let E be a Banach space over a real or complex filed and E is denoted as the Banach dual of E. An element x in the unit sphere SE is called a (real) extre...

‎Let $\Omega_X$ be a bounded‎, ‎circular and strictly convex domain in a complex Banach space $X$‎, ‎and $\mathcal{H}(\Omega_X)$ be the space of all holomorphic functions from $\Omega_X$ to $\mathbb{C}$‎. ‎The growth space $\mathcal{A}^\nu(\Omega_X)$ consists of all $f\in\mathcal{H}(\Omega_X)$‎ ‎such that $$|f(x)|\leqslant C \nu(r_{\Omega_X}(x)),\quad x\in \Omega_X,$$‎ ‎for some constant $C>0$‎...

V. D. Milman proved in [14] that the product of two strictly singular operators on L p [0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying ...

Let $A$ be a Banach algebra, $\Omega(A)$ be the character space of $A$ and $\alpha\in\Omega(A)$. In this paper, we examine the characteristics of $\alpha$-projective (injective) $A$-modules and demonstrate that these character-based $A$-modules also satisfy well-known classical homological properties on Banach $A$-modules.

We prove that every infinite-dimensional Banach space X having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to X \ {0}. More generally, if X is an infinitedimensional Banach space and F is a closed subspace of X such that there is a real-analytic seminorm on X whose set of zeros is F, and X/F is infinite-dimensional, then X and X \ F are real-analytic diffeom...

In this paper, we first show that the iteration {xn} defined by xn+1 = P ((1−αn)xn +αnTP [βnTxn + (1− βn)xn]) converges strongly to some fixed point of T when E is a real uniformly convex Banach space and T is a quasi-nonexpansive non-self mapping satisfying Condition A, which generalizes the result due to Shahzad [11]. Next, we show the strong convergence of the Mann iteration process with err...

We give a brief proof of a recent result of Avron, Seiler and Simon. In [1], it is proved that if P, Q are (not necessarily self-adjoint) projections on a Hilbert space and (P − Q) n is trace-class (i.e. nuclear) for some odd integer n then tr (P − Q) n is an integer and in fact, if P and Q are self-adjoint, tr (P − Q) n = dim E 10 − dim E 01 where E ab = {x : P x = ax, Qx = bx}; (see also [2])...

This paper contains the following results for ecohomies with infinite dimensional commodity spaces. (i) We establish a core-Walras equivalence theorem for economies with an atomless measure space of agents and with an ordered separable Banach commodity space whose positive cone has a non-empty norm interior. This result includes as a special case the Aumann (1964) and Schmeidler-Hildenbrand [Hi...