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, where B L T stands 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, $a=x_{0}<x_{1}<\ldots <x_{{n-1}}<x_{n}=b$ , 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 $\mathbb {R}$ .^[1]

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 B.L.T. 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.

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

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

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