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Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional. Banach–Steinhaus Theorem — Let X {\displaystyle X} be a Banach space and Y {\displaystyle Y} be a normed vector space .
When ′ is the space of finite Radon measures on the real line (so that = is the space of continuous functions vanishing at infinity, by the Riesz representation theorem), the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.
Since every vector v in a Banach space V with a Schauder basis is the limit of P n (v), with P n of finite rank and uniformly bounded, such a space V satisfies the bounded approximation property. A theorem attributed to Mazur [ 6 ] asserts that every infinite-dimensional Banach space V contains a basic sequence, i.e. , there is an infinite ...
If is a Banach space and there exists an invertible bounded compact operator : then is necessarily finite-dimensional. [ 7 ] Now suppose that X {\displaystyle X} is a Banach space and T : X → X {\displaystyle T\colon X\to X} is a compact linear operator, and T ∗ : X ∗ → X ∗ {\displaystyle T^{*}\colon X^{*}\to X^{*}} is the adjoint or ...
In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to infinite dimensions.
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do ...
The Tsirelson space T* is reflexive (Tsirel'son (1974)) and finitely universal, which means that for some constant C ≥ 1, the space T* contains C-isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space X, there exists a subspace Y of the Tsirelson space with multiplicative Banach–Mazur distance to X less than C.
One example for an entirely separable Banach space is the abstract Wiener space construction, similar to a product of Gaussian measures (which are not translation invariant). Another approach is to consider a Lebesgue measure of finite-dimensional subspaces within the larger space and look at prevalent and shy sets. [2]