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A Banach space is a complete normed space (, ‖ ‖) . A normed space is a pair [note 1] (, ‖ ‖) consisting of a vector space over a scalar field (where is commonly or ) together with a distinguished [note 2] norm ‖ ‖: .
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 ]
The bracket , is the scalar product on the Hilbert space; the sum on the right hand side must converge in ... to a Banach space is called ...
In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure, how far a Banach space from a Hilbert space is. The starting point is the Pythagorean identity for orthogonal vectors ( e k ) k = 1 n {\displaystyle (e_{k})_{k=1}^{n}} in Hilbert spaces
The space of sequences has a natural vector space structure by applying scalar addition and multiplication. ... this is a non-separable Banach space which can be seen ...
And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space is locally compact if and only if the unit ball = {: ‖ ‖} is compact, which is the case if and only if is finite-dimensional; this is a consequence of Riesz's lemma. (In fact, a more ...
An important early example was the Banach algebra of essentially bounded measurable functions on a measure space X. This set of functions is a Banach space under pointwise addition and scalar multiplication. With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra.
Non-separable Banach spaces cannot embed isometrically in the separable space C 0 ([0, 1], R), but for every Banach space X, one can find a compact Hausdorff space K and an isometric linear embedding j of X into the space C(K) of scalar continuous functions on K.