Nest algebra

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 $r_1,\dots ,r_n$ are real numbers, $a=x_0<x_1<\dots <x_{n-1}<x_n=b$, and ${\mathit{1}}_S$ denotes the indicator function of the set $S$. The space of all step functions on $[a,b]$, normed by the $L^{\mathrm{\infty }}$ 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 $\mathcal{𝒫}\mathcal{𝒞}$ denote the space of bounded, piecewise continuous functions on $[a,b]$ that are continuous from the right, along with the $L^{\mathrm{\infty }}$ norm. The space $\mathcal{𝒮}$ is dense in $\mathcal{𝒫}\mathcal{𝒞}$, 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 $\mathcal{𝒫}\mathcal{𝒞}$ to $\mathbb{ℝ}$. This defines the Riemann integral of all functions in $\mathcal{𝒫}\mathcal{𝒞}$; for every $f\in \mathcal{𝒫}\mathcal{𝒞}$, $\underset{a}{\overset{b}{\int }}f(x)dx=\tilde{\mathsf{I}}(f)$.

The Hahn–Banach theorem

The above theorem can be used to extend a bounded linear transformation $T:S\to Y$ to a bounded linear transformation from $\overline{S}=X$ 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

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