Editing Vector space (section)

====Banach spaces====
{{Main|Banach space}}
''[[Banach space]]s'', introduced by [[Stefan Banach]], are complete normed vector spaces.{{sfn|Treves|1967|loc=ch. 11}}

A first example is [[Lp space|the vector space <math>\ell^p</math>]] consisting of infinite vectors with real entries <math>\mathbf{x} = \left(x_1, x_2, \ldots, x_n, \ldots\right)</math>
whose [[p-norm|<math>p</math>-norm]] <math>(1 \leq p \leq \infty)</math> given by 
<math display=block>\|\mathbf{x}\|_\infty := \sup_i |x_i| \qquad \text{ for } p = \infty, \text{ and }</math>
<math display=block>\|\mathbf{x}\|_p := \left(\sum_i |x_i|^p\right)^\frac{1}{p} \qquad \text{ for } p < \infty.</math>
<!---- "is finite." - ?!  ---->

The topologies on the infinite-dimensional space <math>\ell^p</math> are inequivalent for different <math>p.</math> For example, the sequence of vectors <math>\mathbf{x}_n = \left(2^{-n}, 2^{-n}, \ldots, 2^{-n}, 0, 0, \ldots\right),</math> in which the first <math>2^n</math> components are <math>2^{-n}</math> and the following ones are <math>0,</math> converges to the [[zero vector]] for <math>p = \infty,</math> but does not for <math>p = 1:</math>
<math display=block>\|\mathbf{x}_n\|_\infty = \sup (2^{-n}, 0) = 2^{-n} \to 0,</math> 
but 
<math display=block>\|\mathbf{x}_n\|_1 = \sum_{i=1}^{2^n} 2^{-n} = 2^n \cdot 2^{-n} = 1.</math>

More generally than sequences of real numbers, functions <math>f : \Omega \to \Reals</math> are endowed with a norm that replaces the above sum by the [[Lebesgue integral]] 
<math display=block>\|f\|_p := \left(\int_{\Omega} |f(x)|^p \, {d\mu(x)}\right)^\frac{1}{p}.</math>

The space of [[integrable function]]s on a given [[domain of a function|domain]] <math>\Omega</math> (for example an interval) satisfying <math>\|f\|_p < \infty,</math> and equipped with this norm are called [[Lp space|Lebesgue spaces]], denoted <math>L^{\;\!p}(\Omega).</math><ref group="nb">The [[triangle inequality]] for <math>\|f + g\|_p \leq \|f\|_p + \|g\|_p</math> is provided by the [[Minkowski inequality]]. For technical reasons, in the context of functions one has to identify functions that agree [[almost everywhere]] to get a norm, and not only a [[seminorm]].</ref>

These spaces are complete.{{sfn|Treves|1967|loc=Theorem 11.2, p. 102}} (If one uses the [[Riemann integral]] instead, the space is {{em|not}} complete, which may be seen as a justification for Lebesgue's integration theory.<ref group="nb">
"Many functions in <math>L^2</math> of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the <math>L^2</math> norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", {{harvtxt|Dudley|1989}}, §5.3, p. 125.</ref>) Concretely this means that for any sequence of Lebesgue-integrable functions <math>f_1, f_2, \ldots, f_n, \ldots</math> with <math>\|f_n\|_p < \infty,</math> satisfying the condition
<math display=block>\lim_{k,\ n \to \infty} \int_{\Omega} \left|f_k(x) - f_n(x)\right|^p \, {d\mu(x)} = 0</math>
there exists a function <math>f(x)</math> belonging to the vector space <math>L^{\;\!p}(\Omega)</math> such that
<math display=block>\lim_{k \to \infty} \int_{\Omega} \left|f(x) - f_k(x)\right|^p \, {d\mu(x)} = 0.</math>

Imposing boundedness conditions not only on the function, but also on its [[derivative]]s leads to [[Sobolev space]]s.{{sfn|Evans|1998|loc = ch. 5}}
{{Clear}}