Editing Lp space (section)

===General ℓ<sup>''p''</sup>-space===

In complete analogy to the preceding definition one can define the space <math>\ell^p(I)</math> over a general [[index set]] <math>I</math> (and <math>1 \leq p < \infty</math>) as
<math display="block">\ell^p(I) = \left\{(x_i)_{i\in I} \in \mathbb{K}^I : \sum_{i \in I} |x_i|^p < +\infty\right\},</math>
where convergence on the right means that only countably many summands are nonzero (see also [[Unconditional convergence]]).
With the norm
<math display="block">\|x\|_p = \left(\sum_{i\in I} |x_i|^p\right)^{1/p}</math>
the space <math>\ell^p(I)</math> becomes a Banach space.
In the case where <math>I</math> is finite with <math> n</math> elements, this construction yields <math>\Reals^n</math> with the <math>p</math>-norm defined above.
If <math>I</math> is countably infinite, this is exactly the sequence space <math>\ell^p</math> defined above.
For uncountable sets <math>I</math> this is a non-[[Separable space|separable]] Banach space which can be seen as the [[Locally convex topological vector space|locally convex]] [[direct limit]] of <math>\ell^p</math>-sequence spaces.<ref>Rafael Dahmen, Gábor Lukács: ''Long colimits of topological groups I: Continuous maps and homeomorphisms.'' in: ''Topology and its Applications'' Nr. 270, 2020. Example 2.14 </ref>

For <math>p = 2,</math> the <math>\|\,\cdot\,\|_2</math>-norm is even induced by a canonical [[inner product]] <math>\langle \,\cdot,\,\cdot\rangle,</math> called the ''{{visible anchor|Euclidean inner product}}'', which means that <math>\|\mathbf{x}\|_2 = \sqrt{\langle\mathbf{x}, \mathbf{x}\rangle}</math> holds for all vectors <math>\mathbf{x}.</math> This inner product can expressed in terms of the norm by using the [[polarization identity]]. 
On <math>\ell^2,</math> it can be defined by
<math display="block">\langle \left(x_i\right)_{i}, \left(y_n\right)_{i} \rangle_{\ell^2} ~=~ \sum_i x_i \overline{y_i}.</math>
Now consider the case <math>p = \infty.</math> Define{{refn|group=note|The condition <math>\sup\operatorname{range} |x| < + \infty.</math> is not equivalent to <math>\sup\operatorname{range} |x|</math> being finite, unless <math>X \neq \varnothing.</math>}}
<math display="block">\ell^\infty(I)=\{x\in \mathbb K^I : \sup\operatorname{range}|x|<+\infty\},</math>
where for all <math>x</math><ref>{{cite book|last1=Garling|first1=D. J. H.|title=Inequalities: A Journey into Linear Analysis|date=2007|publisher=Cambridge University Press|isbn=978-0-521-87624-7|page=54}}</ref>{{refn|group=note|If <math>X = \varnothing</math> then <math>\sup\operatorname{range} |x| = - \infty.</math>}}
<math display="block">\|x\|_\infty\equiv\inf\{C \in \Reals_{\geq 0}:|x_i| \leq C\text{ for all } i \in I\} = \begin{cases}\sup\operatorname{range}|x|&\text{if } X\neq\varnothing,\\0&\text{if } X=\varnothing.\end{cases}</math>

The index set <math>I</math> can be turned into a [[measure space]] by giving it the [[Σ-algebra#Simple set-based examples|discrete σ-algebra]] and the [[counting measure]]. Then the space <math>\ell^p(I)</math> is just a special case of the more general <math>L^p</math>-space (defined below).