Editing Continuous linear extension

{{Short description|Mathematical method in functional analysis}}
In [[functional analysis]], it is often convenient to define a [[linear transformation]] on a [[Complete space|complete]], [[normed vector space]] <math>X</math> by first defining a linear transformation <math>L</math> on a [[dense set|dense]] [[subset]] of <math>X</math> and then [[Continuous extension|continuously extending]] <math>L</math> to the whole space via the theorem below. The resulting extension remains [[Linear map|linear]] and [[Bounded operator|bounded]], and is thus [[continuous function|continuous]], which makes it a '''continuous [[Linear extension (linear algebra)|linear extension]]'''. 

This procedure is known as '''continuous linear extension'''.

==Theorem==

Every bounded linear transformation <math>L</math> from a normed vector space <math>X</math> to a complete, normed vector space <math>Y</math> can be uniquely extended to a bounded linear transformation <math>\widehat{L}</math> from the [[Complete space#Completion|completion]] of  <math>X</math> to <math>Y.</math> In addition, the [[operator norm]] of <math>L</math> is <math>c</math> [[if and only if]] the norm of <math>\widehat{L}</math> is <math>c.</math>

This theorem is sometimes called the '''BLT theorem'''.

==Application==

Consider, for instance, the definition of the [[Riemann integral]]. A [[step function]] on a [[Closure (mathematics)|closed]] [[Interval (mathematics)|interval]] <math>[a,b]</math>  is a function of the form: <math>f\equiv r_1 \mathbf{1}_{[a,x_1)}+r_2 \mathbf{1}_{[x_1,x_2)} + \cdots + r_n \mathbf{1}_{[x_{n-1},b]}</math>
where <math>r_1, \ldots, r_n</math> are real numbers, <math>a = x_0 < x_1 < \ldots < x_{n-1} < x_n = b,</math> and <math>\mathbf{1}_S</math> denotes the [[indicator function]] of the set <math>S.</math> The space of all step functions on <math>[a,b],</math> normed by the <math>L^\infty</math> norm (see [[Lp space]]), is a normed vector space which we denote by <math>\mathcal{S}.</math> Define the integral of a step function by: <math display=block>I \left(\sum_{i=1}^n r_i \mathbf{1}_{ [x_{i-1},x_i)}\right) = \sum_{i=1}^n r_i (x_i-x_{i-1}).</math> 
<math>I</math> as a function is a bounded linear transformation from <math>\mathcal{S}</math> into <math>\R.</math><ref> Here, <math>\R</math> is also a normed vector space; <math>\R</math> is a vector space because it satisfies all of the [[Vector space#Formal definition|vector space axioms]] and is normed by the [[Absolute value|absolute value function]].</ref>

Let <math>\mathcal{PC}</math> denote the space of bounded, [[piecewise]] continuous functions on <math>[a,b]</math> that are continuous from the right, along with the <math>L^\infty</math> norm. The space <math>\mathcal{S}</math> is dense in <math>\mathcal{PC},</math> so we can apply the BLT theorem to extend the linear transformation <math>I</math> to a bounded linear transformation <math>\widehat{I}</math> from <math>\mathcal{PC}</math> to <math>\R.</math> This defines the Riemann integral of all functions in <math>\mathcal{PC}</math>; for every <math>f\in \mathcal{PC},</math> <math>\int_a^b f(x)dx=\widehat{I}(f).</math>

==The Hahn–Banach theorem==

The above theorem can be used to extend a bounded linear transformation <math>T : S \to Y</math> to a bounded linear transformation from <math>\bar{S} = X</math> to <math>Y,</math> ''if'' <math>S</math> is dense in <math>X.</math> If <math>S</math> is not dense in <math>X,</math> then the [[Hahn–Banach theorem]] may sometimes be used to show that an extension [[existence|exists]]. However, the extension may not be unique.

==See also==

* {{annotated link|Closed graph theorem (functional analysis)}}
* {{annotated link|Continuous linear operator}}
* {{annotated link|Densely defined operator}}
* {{annotated link|Hahn–Banach theorem}}
* {{annotated link|Linear extension (linear algebra)}}
* {{annotated link|Partial function}}
* {{annotated link|Vector-valued Hahn–Banach theorems}}

==References==

{{reflist}}

* {{cite book|last=Reed|first=Michael|author2=Barry Simon|year=1980|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis|publisher=Academic Press|location=San Diego|isbn=0-12-585050-6}}

{{Banach spaces}}
{{Functional analysis}}
{{Topological vector spaces}}

{{DEFAULTSORT:Continuous Linear Extension}}

[[Category:Functional analysis]]
[[Category:Linear operators]]