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. [3] Banás , J. , Goebel , K. 1980 Measures of Noncompactness in Banach Spaces Lecture Notes in Pure and Applied Mathemathics 60 Marcel Dekker New York . [4

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between two Banach spaces, Electron. J. Diff. Equ. , Vol. 2014 (2014), no. 99, pp. 1–6. Schmeidel E. Conditions for having a diffeomorphis between two Banach spaces

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Abstract  

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case.

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. [6] Jain , P. K. , Ahmad , K. 1981 Certain characterisations of Schauder decompositions in Banach spaces Analele Mathematica 19 61 – 66 . [7] Marti , J. T

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Abstract  

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and fL p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result.

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Abstract  

We define two properties of sequences in Banach spaces that may be related to measures of noncompactness of subsets of these spaces. The first one concerns properties of sequences related to the strong topology, and the second one is related to the weak topology. Given a Banach space X, we introduce a new Banach space such that we can find a subset E in it that may be identified with the balls in the first one. We use compactness in this new space to characterize our sequential properties. In particular, we prove a general form of the Eberlein-Smulian theorem.

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Given a sequence of identically distributed ψ-mixing random variables {X n; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.

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The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.

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Given a complex Banach space X and a holomorphic function f on its unit ball B, we discuss the problem whether f can be approximated, uniformly on smaller balls, by functions g holomorphic on all of X.

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We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : CC has a fixed point. As its application, we give an existence result for the solution of an integral equation.

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