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

Gowers, W. T.

Bull. Lond. Math. Soc. 26, No. 6, 523-530 (1994).

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