| dbp:mathStatement
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- Let be a linear mapping between Banach spaces. The graph of is closed in if and only if is continuous. (en)
- If a Banach space is the internal direct sum of closed subspaces then is isomorphic to (en)
- Let be a normed vector space. Then the closed unit ball of the dual space is compact in the weak* topology. (en)
- If and are compact Hausdorff spaces and if and are isometrically isomorphic, then the topological spaces and are homeomorphic. (en)
- For every measure the space is weakly sequentially complete. (en)
- Let be a separable Banach space. The following are equivalent:
*The space does not contain a closed subspace isomorphic to
*Every element of the bidual is the weak*-limit of a sequence in (en)
- A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space. (en)
- for all (en)
- Let be a normed space. If is separable, then is separable. (en)
- Suppose that and are Banach spaces and that Suppose further that the range of is closed in Then is isomorphic to (en)
- Let be a bounded sequence in a Banach space. Either has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of (en)
- Let be a Banach space and be a normed vector space. Suppose that is a collection of continuous linear operators from to The uniform boundedness principle states that if for all in we have then (en)
- Let be an uncountable compact metric space. Then is isomorphic to (en)
- Let be a vector space over the field Let further
* be a linear subspace,
* be a sublinear function and
* be a linear functional so that for all
Then, there exists a linear functional so that (en)
- For a Banach space the following two properties are equivalent:
* is reflexive.
* for all in there exists with so that (en)
- A set in a Banach space is relatively weakly compact if and only if every sequence in has a weakly convergent subsequence. (en)
- Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism. (en)
- Let and be Banach spaces and be a surjective continuous linear operator, then is an open map. (en)
- Let be a reflexive Banach space. Then is separable if and only if is separable. (en)
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