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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.

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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