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 $\mathsf$ on a dense subset of $X$ and then extending $\mathsf$ 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 $\mathsf$ from a normed vector space $X$ to a complete, normed vector space $Y$ can be uniquely extended to a bounded linear transformation $\tilde$ from the completion of $X$ to $Y.$ In addition, the operator norm of $\mathsf$ is $c$ if and only if the norm of $\tilde$ is $c.$

This theorem is sometimes called the BLT theorem.

Application

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval ,b/math> is a function of the form: $f\equiv r_1 \mathit_+r_2 \mathit_ + \cdots + r_n \mathit_$
where $r_1, \ldots, r_n$ are real numbers, $a = x_0 < x_1 < \ldots < x_ < x_n = b,$ and $\mathit_S$ denotes the indicator function of the set $S.$ The space of all step functions on ,b normed by the $L^\infty$ norm (see Lp space), is a normed vector space which we denote by $\mathcal.$ Define the integral of a step function by: $\mathsf \left(\sum_^n r_i \mathit_\right) = \sum_^n r_i (x_i-x_).$
$\mathsf$ as a function is a bounded linear transformation from $\mathcal$ into $\R.$ Here, $\R$ is also a normed vector space; $\R$ is a vector space because it satisfies all of the vector space axioms and is normed by the absolute value function.

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

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 $\bar = X$ 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.

See also

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References

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Footnotes

{{DEFAULTSORT:Continuous Linear Extension
Functional analysis