Search results
Results from the WOW.Com Content Network
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.
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 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 normed vector space ((,), ‖ ‖) is called space or the Lebesgue space of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem).
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.
A bounded disk in a topological vector space such that (,) is a Banach space is called a Banach disk, infracomplete, or a bounded completant in . If its shown that ( span D , p D ) {\displaystyle \left(\operatorname {span} D,p_{D}\right)} is a Banach space then D {\displaystyle D} will be a Banach disk in any TVS that contains D ...
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 ...