Continuous linear extension

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space ${\displaystyle X}$ by first defining a linear transformation ${\displaystyle {\mathsf {T}}}$ on a dense subset of ${\displaystyle X}$ and then extending ${\displaystyle {\mathsf {T}}}$ to the whole space via the theorem below. The resulting extension remains linear and bounded (thus continuous).

This procedure is known as continuous linear extension.

Theorem

Every bounded linear transformation ${\displaystyle {\mathsf {T}}}$  from a normed vector space ${\displaystyle X}$  to a complete, normed vector space ${\displaystyle Y}$  can be uniquely extended to a bounded linear transformation ${\displaystyle {\tilde {\mathsf {T}}}}$  from the completion of ${\displaystyle X}$  to ${\displaystyle Y}$ . In addition, the operator norm of ${\displaystyle {\mathsf {T}}}$  is ${\displaystyle c}$  iff the norm of ${\displaystyle {\tilde {\mathsf {T}}}}$  is ${\displaystyle c}$ .

This theorem is sometimes called the B L T theorem, for bounded linear transformation.

Application

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval ${\displaystyle [a,b]}$  is a function of the form: ${\displaystyle f\equiv r_{1}{\mathit {1}}_{[a,x_{1})}+r_{2}{\mathit {1}}_{[x_{1},x_{2})}+\cdots +r_{n}{\mathit {1}}_{[x_{n-1},b]}}$  where ${\displaystyle r_{1},\ldots ,r_{n}}$  are real numbers, ${\displaystyle a=x_{0} , and ${\displaystyle {\mathit {1}}_{S}}$  denotes the indicator function of the set ${\displaystyle S}$ . The space of all step functions on ${\displaystyle [a,b]}$ , normed by the ${\displaystyle L^{\infty }}$  norm (see Lp space), is a normed vector space which we denote by ${\displaystyle {\mathcal {S}}}$ . Define the integral of a step function by: ${\displaystyle {\mathsf {I}}\left(\sum _{i=1}^{n}r_{i}{\mathit {1}}_{[x_{i-1},x_{i})}\right)=\sum _{i=1}^{n}r_{i}(x_{i}-x_{i-1})}$ . ${\displaystyle {\mathsf {I}}}$  as a function is a bounded linear transformation from ${\displaystyle {\mathcal {S}}}$  into ${\displaystyle \mathbb {R} }$ .[1]

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

The Hahn–Banach theorem

The above theorem can be used to extend a bounded linear transformation ${\displaystyle T:S\rightarrow Y}$  to a bounded linear transformation from ${\displaystyle {\bar {S}}=X}$  to ${\displaystyle Y}$ , if ${\displaystyle S}$  is dense in ${\displaystyle X}$ . If ${\displaystyle S}$  is not dense in ${\displaystyle X}$ , then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.

References

• Reed, Michael; Barry Simon (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. San Diego: Academic Press. ISBN 0-12-585050-6.

Footnotes

1. ^ Here, ${\displaystyle \mathbb {R} }$  is also a normed vector space; ${\displaystyle \mathbb {R} }$  is a vector space because it satisfies all of the vector space axioms and is normed by the absolute value function.