Much of his early work focused on proofs surrounding Hilbert space and Hilbert cubes.
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Conversely, every Polish space is homeomorphic to a G ?-subset of the Hilbert cube.
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The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.
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In the latter case, the resulting topology is the box topology; cylinder sets are never Hilbert cubes.
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Any such sequence belongs to the Hilbert space ! 2, so the Hilbert cube inherits a metric from there.
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The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval [ 0, 1 ].
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Compact sets in Banach spaces may also carry natural measures : the Hilbert cube, for instance, carries the product Lebesgue measure.
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Every G ?-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and complete metric space.
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Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube ( see below ).
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Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube ( see below ).
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