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, $a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b$, 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 $\mathbb {R}$.

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.