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Until the 1970s, one of the principal problems of the theory of Banach spaces was the basis problem dealt with by Banach himself: Does a basis exist in each separable Banach space? The question of existence of a basis in specifically defined Banach spaces remained open as well. Each $T$-basis is a complete minimal (not necessarily uniformly minimal) system with a total adjoint.
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The theory of Banach spaces is a thoroughly studied branch of functional analysis, with numerous applications in various branches of mathematics - directly or by way of the theory of operators.Ī Banach space $X$ is a vector space over $\R$ or $\C$ with a norm $\norm^\infty$ in $C$ is a summation basis for the methods of Cesàro and Abel. The ideas of weak convergence of elements and linear functionals in Banach spaces ultimately evolved to the concept of weak topology. Thus, the idea of semi-norms, taken from the theory of normed spaces, became an indispensable tool in constructing the theory of locally convex linear topological spaces. These theories mutually enriched one another with new ideas and facts. The theory of Banach spaces developed in parallel with the general theory of linear topological spaces.
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Banach who in 1922 began a systematic study of these spaces, based on axioms introduced by himself, and who obtained highly advanced results. It is in these spaces that the fundamental concepts of strong and weak convergence, compactness, linear functional, linear operator, etc., were originally studied. Riesz between 19 served as the starting point for the theory of Banach spaces.