| 41. | This means that the correspondence defines a linear operator between the Banach spaces and.
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| 42. | If we take a Banach space completion, it becomes a C * algebra.
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| 43. | A further generalisation is to Fr�chet manifolds, replacing Banach spaces by Fr�chet spaces.
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| 44. | Turning it into a normed vector space ( in fact a Banach space ).
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| 45. | Let be a compatible couple of Banach spaces.
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| 46. | As a complete normed space, Hilbert spaces are by definition also Banach spaces.
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| 47. | Note, this theorem fails for complex Banach spaces
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| 48. | When is a Banach space, it is viewed as a closed linear subspace of.
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| 49. | Indeed, if the dual of a Banach space is separable, then is separable.
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| 50. | Finally, all of the above holds for integrals with values in a Banach space.
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