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By Vinogradov S.S., et al.

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1) does not ensure convergence, then analysis in such a space is likely to present serious difficulties. Attention will therefore usually be restricted from here on to spaces in which all Cauchy sequences are convergent. 4 Definition. A set S in a normed vector space 1/ is said to be complete iff each Cauchy sequence in S converges to a point of S. 1/ itself is known as a complete normed vector space or a Banach space iff it is complete. Throughout the symbols fJB and C(j will always denote Banach spaces.

K' .... 4 21 BANACH SPACES A Cauchy sequence (f) in a set S may be regarded as "potentially convergent" in that its terms get closer together as n ~ 00, a property possessed by convergent sequences. Whether or not the sequence fulfills its potentiality depends roughly on whether S is "large enough" or complete in the present terminology. Consider for example the subset S = (0, 1] of [f;t The sequence (n - 1) is evidently Cauchy, but its limit in [R is which does not lie in S. Sis therefore not complete.

If fJ6 is a Banach space, it is a Banach space in any equivalent norm. The proof is easy and is left as an exercise. In a finite dimensional space all norms are equivalent, and every finite dimensional normed vector space is a Banach space. 11. 20 Example. I] whence we deduce that 11'11* and the sup norm are equivalent on ,(;([O, 1]). 15. Our final remarks concern bases in Banach spaces. 2 was for finite dimensional spaces only, and obviously needs modification if it is to be applied in a general context.

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