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In mathematics, more specifically in functional analysis, a Banach space (/ ˈ b ɑː. n ʌ x /, Polish pronunciation:) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is ...
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. Function f is almost surely separably valued (or essentially separably valued ) if there exists a subset N ⊆ X with μ ( N ) = 0 such that f ( X \ N ) ⊆ B is separable.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
i.e. the space of finitely additive signed measures on Σ that are absolutely continuous with respect to μ (μ-a.c. for short). When the measure space is furthermore sigma-finite then L ∞ (μ) is in turn dual to L 1 (μ), which by the Radon–Nikodym theorem is identified with the set of all countably additive μ-a.c. measures. In other ...
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires .
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
Theorem — Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. Every nonzero λ ∈ σ(C) is an eigenvalue of C. For all nonzero λ ∈ σ(C), there exist m such that Ker((λ − C) m) = Ker((λ − C) m+1), and this subspace is finite-dimensional. The eigenvalues can only accumulate at 0.
The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanisław Mazur (left) in 1936. [ 1 ] In mathematics , specifically functional analysis , a Banach space is said to have the approximation property (AP) , if every compact operator is a limit of finite-rank ...