Editing Lp space (section)

===Special cases===
For <math>1 \leq p \leq \infty</math> the <math>\ell^p</math> spaces are a special case of <math>L^p</math> spaces; when <math>S</math> are the [[natural number]]s <math>\mathbb{N}</math> and <math>\mu</math> is the [[counting measure]]. More generally, if one considers any set <math>S</math> with the counting measure, the resulting <math>L^p</math> space is denoted <math>\ell^p(S).</math> For example, <math>\ell^p(\mathbb{Z})</math> is the space of all sequences indexed by the integers, and when defining the <math>p</math>-norm on such a space, one sums over all the integers. The space <math>\ell^p(n),</math> where <math>n</math> is the set with <math>n</math> elements, is <math>\Reals^n</math> with its <math>p</math>-norm as defined above. 

Similar to <math>\ell^2</math> spaces, <math>L^2</math> is the only [[Hilbert space]] among <math>L^p</math> spaces. In the complex case, the inner product on <math>L^2</math> is defined by
<math display="block">\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x).</math>
Functions in <math>L^2</math> are sometimes called '''[[square-integrable function]]s''', '''quadratically integrable functions''' or '''square-summable functions''', but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a [[Riemann integral]] {{harv|Titchmarsh|1976}}.

As any Hilbert space, every space <math>L^2</math> is linearly isometric to a suitable <math>\ell^2(I),</math> where the cardinality of the set <math>I</math> is the cardinality of an arbitrary basis for this particular <math>L^2.</math> 

If we use complex-valued functions, the space <math>L^\infty</math> is a [[commutative]] [[C*-algebra]] with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative [[von Neumann algebra]]. An element of <math>L^\infty</math> defines a [[bounded operator]] on any <math>L^p</math> space by [[multiplication operator|multiplication]].