# Continuous linear extension

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space $X$ by first defining a linear transformation ${\mathsf {T}}$ on a dense subset of $X$ and then extending ${\mathsf {T}}$ 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 ${\mathsf {T}}$  from a normed vector space $X$  to a complete, normed vector space $Y$  can be uniquely extended to a bounded linear transformation ${\tilde {\mathsf {T}}}$  from the completion of $X$  to $Y$ . In addition, the operator norm of ${\mathsf {T}}$  is $c$  iff the norm of ${\tilde {\mathsf {T}}}$  is $c$ .

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

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

## The Hahn–Banach theorem

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