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Banach Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
This chapter deals mainly with the studies of Banach spaces. It contains the most important examples of Banach spaces and results on the bounded linear operators between normed and Banach spaces. The first highlight is the fundamental Hahn–Banach extension theorem, Theorem 4.5.2, which is of vital importance in the proof of many results in functional analysis; also many of its corollaries are given; it can be found in [10, Théorème 1, Théorème 2]. Furthermore, the versions of Banach spaces are given in Section 4.6 of the open mapping and closed graph theorems, Banach’s theorem of the bounded inverse, and the Banach–Steinhaus theorem, an alternative proof of which is added that does not use Baire’s theorem. Two important applications of the closed graph theorem are the criterion for a closed subspace in a Banach space to have a topological complement, and the Eni–Karauš theorem, Theorem 4.6.13. Further applications of the Hahn–Banach and Banach–Steinhaus theorems are the classical representation theorems for the continuous linear functionals on the classical sequence spaces, including the representation theorem in the case of the space of bounded real sequences, and the Riesz representation theorem, Theorem 4.7.7, for the continuous linear functionals on the space of continuous real functions on the unit interval. The other topics focus on the reflexivity of spaces, the studies of adjoint operators, quotient spaces, Cauchy nets, the equivalence of norms, compactness and the Riesz lemma, Lemma 4.13.2, compact operators and operators with closed range.
Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
Let V be a vector space over field F. Then a linear transformation of the form f : V → F, is referred to as a linear functional. The collection of all linear functionals defined on vector space V is a vector space called the dual space of V and is represented in terms of the notation V*. The following result establishes a procedure for constructing a basis for the dual space [6].
Operators
Published in Sandeep Kumar, Ashish Pathak, Debashis Khan, Mathematical Theory of Subdivision, 2019
Sandeep Kumar, Ashish Pathak, Debashis Khan
Linear functional is a special case of linear operator where the operator assigns scalars to vectors. In other words, V=ℝ.
Blow-up time analysis of parabolic equations with variable nonlinearities
Published in Applicable Analysis, 2022
Benkouider Soufiane, Rahmoune Abita
We say that for a function, is a weak solution of problem (1) if for every test-function and every , the following identity holds: where is the dual space of (the space of linear functionals over ). For every fixed we introduce the Banach space with the associate norm and denote by its dual.
Optimal economic growth problems with high values of total factor productivity
Published in Applicable Analysis, 2022
To deal with the state constraint in , one introduces a multiplier that is an element in the topological dual of the space of continuous functions with the supremum norm. By the Riesz Representation Theorem (see, e.g.[22, Theorem 6, p. 374] and [23, Theorem 1, p. 113–115]), any bounded linear functional f on can be uniquely represented in the form where v is a function of bounded variation on which vanishes at and which are continuous from the right at every point , and is the Riemann–Stieltjes integral of x with respect to v (see, e.g.[22, p. 364]). The set of the elements of which are given by nondecreasing functions v is denoted by . The integrals and of a Borel measurable function ν in next theorem are understood in the sense of the Lebesgue–Stieltjes integration [22, p. 364].
Total asymptotically nonexpansive mappings and generalized variational-like inclusion problems in semi-inner product spaces
Published in Optimization, 2022
Javad Balooee, Suliman Al-Homidan
By a careful reading the proof of Proposition 1 (i) in [71], we note that a well-known corollary of the Hahn-Banach theorem is used to derive the desired result. In the light of the normed space version of the Hahn-Banach extension theorem, any continuous linear functional defined on any subspace of a normed space has at least one Hahn-Banach extension. Indeed, this extension is not necessarily unique, and there can even be infinitely. Thanks to this fact and Lemma 2.6 (i) it follows that for any , in general, contains infinitely many different elements of .