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Let (,,) be a measure space, and be a Banach space.The Bochner integral of a function : is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form = = (), where the are disjoint members of the -algebra , the are distinct elements of , and χ E is the characteristic function of .
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 within the space.
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .
Let E be a Banach space such that both E and its continuous dual space E ∗ are separable spaces; let μ be a Borel measure on E. Let S be any (fixed) subset of the class of functions defined on E. A linear operator A : S → L 2 (E, μ; R) is said to be an integration by parts operator for μ if
Tsirelson space, a reflexive Banach space in which neither nor can be embedded. W.T. Gowers construction of a space X {\displaystyle X} that is isomorphic to X ⊕ X ⊕ X {\displaystyle X\oplus X\oplus X} but not X ⊕ X {\displaystyle X\oplus X} serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem [ 1 ]
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm.
There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions (() = ()). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions ...
Note the special case where both Hilbert spaces are identical and is an operator on that Hilbert space. When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, of an operator :, where , are Banach spaces with corresponding norms ‖ ‖, ‖ ‖.