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A solution to Banach’s hyperplane problem. (English) Zbl 0838.46011

Summary: An infinite-dimensional Banach space \(X\) is constructed which is not isomorphic to \(X\oplus \mathbb{R}\). Equivalently, \(X\) is not isomorphic to any of its closed subspaces of codimension one. This gives a negative answer to a question of Banach. In fact, \(X\) has the stronger property that it is not isomorphic to any proper subspace. It also happens to have an unconditional basis.

MSC:

46B20 Geometry and structure of normed linear spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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