| Banach space A . Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every converges to an element of the space. All finite-dimensional and normed vector spaces are complete and thus are are Banach spaces. Using absolute value for the norm, the real numbers are a Banach space whereas the rationals are not. This is because there are sequences of rationals that converges to irrationals. Several theorems hold only in Banach spaces, e.g. the . All finite-dimensional real and complex vector spaces are Banach spaces. , spaces of , and spaces of are examples of infinite-dimensional Banach spaces. Applications include , , and radar. [Robert E. Megginson, "An Introduction to Banach Space Theory", Graduate Texts in Mathematics, 183, Springer Verlag, September 1998]. (2000-03-10) |
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