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→Banach spaces: the positivity is implicit? |
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Examples of Banach spaces are [[Lp space|<math>L^p</math>-spaces]] for any real number {{nowrap|<math>p\geq1</math>.}} Given also a measure <math>\mu</math> on set {{nowrap|<math>X</math>,}} then {{nowrap|<math>L^p(X)</math>,}} sometimes also denoted <math>L^p(X,\mu)</math> or {{nowrap|<math>L^p(\mu)</math>,}} has as its vectors equivalence classes <math>[\,f\,]</math> of [[Lebesgue-measurable function|measurable function]]s whose [[absolute value]]'s <math>p</math>-th power has finite integral; that is, functions <math>f</math> for which one has
<math display="block">\int_{X}\left|f(x)\right|^p\,d\mu(x) <
If <math>\mu</math> is the [[counting measure]], then the integral may be replaced by a sum. That is, we require
<math display="block">\sum_{x\in X}\left|f(x)\right|^p <
Then it is not necessary to deal with equivalence classes, and the space is denoted {{nowrap|<math>\ell^p(X)</math>,}} written more simply <math>\ell^p</math> in the case when <math>X</math> is the set of non-negative [[integer]]s.
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