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 $L$ on a dense subset of $X$ and then continuously extending $L$ to the whole space via the theorem below. The resulting extension remains linear and bounded, and is thus continuous, which makes it a continuous linear extension.

This procedure is known as continuous linear extension.

Theorem

Every bounded linear transformation $L$ from a normed vector space $X$ to a complete, normed vector space $Y$ can be uniquely extended to a bounded linear transformation ${\widehat {L}}$ from the completion of $X$ to $Y.$ In addition, the operator norm of $L$ is $c$ if and only if the norm of ${\widehat {L}}$ 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 $[a,b]$ is a function of the form: $f\equiv r_{1}\mathbf {1} _{[a,x_{1})}+r_{2}\mathbf {1} _{[x_{1},x_{2})}+\cdots +r_{n}\mathbf {1} _{[x_{n-1},b]}$ where $r_{1},\ldots ,r_{n}$ are real numbers, $a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b,$ and $\mathbf {1} _{S}$ 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: $I\left(\sum _{i=1}^{n}r_{i}\mathbf {1} _{[x_{i-1},x_{i})}\right)=\sum _{i=1}^{n}r_{i}(x_{i}-x_{i-1}).$ $I$ 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 BLT theorem to extend the linear transformation $I$ to a bounded linear transformation ${\widehat {I}}$ from ${\mathcal {PC}}$ to $\mathbb {R} .$ This defines the Riemann integral of all functions in ${\mathcal {PC}}$ ; for every $f\in {\mathcal {PC}},$ $\int _{a}^{b}f(x)dx={\widehat {I}}(f).$

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 ${\bar {S}}=X$ 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

^ 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.

Reed, Michael; Barry Simon (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. San Diego: Academic Press. ISBN 0-12-585050-6.

[1] 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.

[1]

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Category

Theorem

Application

The Hahn–Banach theorem

See also

References