Editing Lp space (section)

=={{math|''ℓ''{{i sup|''p''}}}} spaces and sequence spaces==
{{Details|Sequence space}}
The <math>p</math>-norm can be extended to vectors that have an infinite number of components ([[sequence]]s), which yields the space <math>\ell^p.</math> This contains as special cases:
* <math>\ell^1,</math> the space of sequences whose series are [[absolute convergence|absolutely convergent]],
* <math>\ell^2,</math> the space of '''square-summable''' sequences, which is a [[Hilbert space]], and
* <math>\ell^\infty,</math> the space of [[bounded sequence]]s.

The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinite [[sequence]]s of real (or [[complex number|complex]]) numbers are given by:
<math display="block">\begin{align}
& (x_1, x_2, \ldots, x_n, x_{n+1},\ldots)+(y_1, y_2, \ldots, y_n, y_{n+1},\ldots) \\
= {} & (x_1+y_1, x_2+y_2, \ldots, x_n+y_n, x_{n+1}+y_{n+1},\ldots), \\[6pt]
& \lambda \cdot \left (x_1, x_2, \ldots, x_n, x_{n+1},\ldots \right) \\
= {} & (\lambda x_1, \lambda x_2, \ldots, \lambda x_n, \lambda x_{n+1},\ldots).
\end{align}</math>

Define the <math>p</math>-norm:
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2|^p + \cdots +|x_n|^p + |x_{n+1}|^p + \cdots\right)^{1/p}</math>

Here, a complication arises, namely that the [[series (mathematics)|series]] on the right is not always convergent, so for example, the sequence made up of only ones, <math>(1, 1, 1, \ldots),</math> will have an infinite <math>p</math>-norm for <math>1 \leq p < \infty.</math> The space <math>\ell^p</math> is then defined as the set of all infinite sequences of real (or complex) numbers such that the <math>p</math>-norm is finite.

One can check that as <math>p</math> increases, the set <math>\ell^p</math> grows larger. For example, the sequence
<math display="block">\left(1, \frac{1}{2}, \ldots, \frac{1}{n}, \frac{1}{n+1}, \ldots\right)</math>
is not in <math>\ell^1,</math> but it is in <math>\ell^p</math> for <math>p > 1,</math> as the series
<math display="block">1^p + \frac{1}{2^p} + \cdots + \frac{1}{n^p} + \frac{1}{(n+1)^p} + \cdots,</math>
diverges for <math>p = 1</math> (the [[harmonic series (mathematics)|harmonic series]]), but is convergent for <math>p > 1.</math>

One also defines the <math>\infty</math>-norm using the [[supremum]]:
<math display="block">\|x\|_\infty = \sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \ldots)</math>
and the corresponding space <math>\ell^\infty</math> of all bounded sequences. It turns out that<ref>{{Citation| last1=Maddox | first1=I. J. | title=Elements of Functional Analysis | publisher=CUP | location=Cambridge | edition=2nd | year=1988}}, page 16</ref>
<math display="block">\|x\|_\infty = \lim_{p \to \infty} \|x\|_p</math>
if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider <math>\ell^p</math> spaces for <math>1 \leq p \leq \infty.</math>

The <math>p</math>-norm thus defined on <math>\ell^p</math> is indeed a norm, and <math>\ell^p</math> together with this norm is a [[Banach space]].

===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).