Banach space properties
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The statement of the theorem
Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuoussurjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces. The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set. In the case of Hilbert spaces, this is basic in the study of unbounded operators (see Closed operator).
The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.
- ^Halmos 1982, Problem 52, 58
- ^Rudin 1973
- ^Trèves 1967, Chapter 18