Theorems and Properties
- Theorem A normed space X is a Banach space if and only if each absolutely convergent series in X converges.
- Theorem Let X be a normed space. Then there is a Banach space Y and an isometric isomorphism T: X → Y such that T(X) is dense in Y. Furthermore, the space X′ is isometrically isomorphic to Y′. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z, then Z is isometrically isomorphic to Y.
- Theorem Let X and Y be normed spaces. Then, B(X, Y) := {T: X → Y | T linear and bounded}, is a normed space under the operator norm. If Y is a Banach space, then so is B(X, Y).
- Proposition Let T be a linear operator from a normed space X into a normed space Y. If X is a Banach space and T is an isomorphism, then T(X) is a Banach space.
- Corollary Every finite-dimensional normed space is a Banach space.
- Corollary A Banach space with a countable Hamel basis is finite-dimensional.
- The Open Mapping Theorem Let X and Y be Banach spaces and T: X → Y be a continuous linear operator. Then T is surjective if and only if T is an open map. In particular, if T is bijective and continuous, then T−1 is also continuous.
- Corollary Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.
- The Closed Graph Theorem Let T: X → Y be a linear function between Banach spaces. The graph is closed in X × Y if and only if T is continuous.
- Theorem If M is a closed subspace of a Banach space X, then X/M with is also a Banach space.
- The First Isomorphism theorem for Banach spaces Suppose that X and Y are Banach spaces and that T ∈ B(X, Y). Suppose further that the range of T is closed in Y. Then X/Ker(T) ≅ T(X). This is a topological isomorphism with means that a bijective, linear set L exists which goes from X/Ker(T) to T(X) so that both L and L−1 are continuous.
- Theorem Let X1, ..., Xn be normed spaces. Then X1 ⊕ ... ⊕ Xn is a Banach space if and only if each Xj is a Banach space.
- Proposition If X is a Banach space that is the internal direct sum of its closed subspaces M1, ..., Mn, then X ≅ M1 ⊕ ... ⊕ Mn.
- Theorem Every Banach space is a Fréchet space.
- Theorem For every separable Banach space X, there is a closed subspace M of ℓ1 such that X ≅ ℓ1/M.
- Hahn–Banach theorem Let X be a vector space of the field K. Let further
- Y ⊆ X be a linear subspace,
- p: X → R be a sublinear function and
- f: Y → K be a linear functional so that Re f(y) ≤ p(y) for all y in Y.
Then, there exists a linear functional F: X → K so that - and
- .
In particular, every continuous linear functional on a subspace of a normed space can continuously be continued on the whole space.
- Banach–Steinhaus theorem Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. The uniform boundedness principle states that if for all x in X we have, then
- Banach–Alaoglu theorem Let X be a normed vector space. Then the closed unit ball of the dual space B′ := {x ∈ X′ | ǁxǁ ≤ 1} is compact in the weak* topology.
Read more about this topic: Banach Space
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
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