| 31. | Here c _ 0 is the Banach space of sequences converging to zero.
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| 32. | Suppose that and are Banach spaces and that.
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| 33. | For every separable Banach space, there is a closed subspace of such that.
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| 34. | Let be a linear mapping between Banach spaces.
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| 35. | Precisely, for every Banach space, the map
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| 36. | A Banach space isomorphic to is homogeneous, and Banach asked for the converse.
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| 37. | Hilbert spaces, Banach spaces or Fr�chet spaces.
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| 38. | Leonard Gross provided the generalization to the case of a general separable Banach space.
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| 39. | They are generalizations of Banach spaces ( normed vector spaces that are norm ).
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| 40. | In contrast to Banach spaces, the metric need not arise from a norm.
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