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- A continuous bijective linear operator between Banach spaces has continuous inverse. That is, the inverse operator is continuous. (en)
- Let and be two F-spaces. Then every continuous linear map of onto is a TVS homomorphism,
where a linear map is a topological vector space homomorphism if the induced map is a TVS-isomorphism onto its image. (en)
- Let be a surjective continuous linear map between Banach spaces . Then is an open mapping . (en)
- Let be a continuous linear map between normed spaces.
If is nearly-open and if is complete, then is open and surjective.
More precisely, if for some and if is complete, then
:
where is an open ball with radius and center . (en)
- Let and be Banach spaces, let and denote their open unit balls, and let be a bounded linear operator.
If then among the following four statements we have
# for all = continuous dual of ;
# ;
# ;
# is surjective.
Furthermore, if is surjective then holds for some (en)
- Let be a continuous linear operator from a complete pseudometrizable TVS onto a Hausdorff TVS
If is nonmeager in then is a open map and is a complete pseudometrizable TVS.
Moreover, if is assumed to be hausdorff , then is also an F-space. (en)
- If is a continuous linear bijection from a complete Pseudometrizable topological vector space onto a Hausdorff TVS that is a Baire space, then is a homeomorphism . (en)
- Let be a surjective linear map from a complete pseudometrizable TVS onto a TVS and suppose that at least one of the following two conditions is satisfied:
# is a Baire space, or
# is locally convex and is a barrelled space,
If is a closed linear operator then is an open mapping.
If is a continuous linear operator and is Hausdorff then is an open mapping. (en)
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