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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
##### Keywords:
unconditional basis
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