Continuous linear extension

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space ${\displaystyle X}$ by first defining a linear transformation ${\displaystyle {\mathsf {T}}}$ on a dense subset of ${\displaystyle X}$ and then extending ${\displaystyle {\mathsf {T}}}$ 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 ${\displaystyle {\mathsf {T}}}$ from a normed vector space ${\displaystyle X}$ to a complete, normed vector space ${\displaystyle Y}$ can be uniquely extended to a bounded linear transformation ${\displaystyle {\tilde {\mathsf {T}}}}$ from the completion of ${\displaystyle X}$ to ${\displaystyle Y}$. In addition, the operator norm of ${\displaystyle {\mathsf {T}}}$ is ${\displaystyle c}$ iff the norm of ${\displaystyle {\tilde {\mathsf {T}}}}$ is ${\displaystyle c}$.

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 ${\displaystyle [a,b]}$ is a function of the form: ${\displaystyle f\equiv r_{1}{\mathit {1}}_{[a,x_{1})}+r_{2}{\mathit {1}}_{[x_{1},x_{2})}+\cdots +r_{n}{\mathit {1}}_{[x_{n-1},b]}}$ where ${\displaystyle r_{1},\ldots ,r_{n}}$ are real numbers, ${\displaystyle a=x_{0}, and ${\displaystyle {\mathit {1}}_{S}}$ denotes the indicator function of the set ${\displaystyle S}$. The space of all step functions on ${\displaystyle [a,b]}$, normed by the ${\displaystyle L^{\infty }}$ norm (see Lp space), is a normed vector space which we denote by ${\displaystyle {\mathcal {S}}}$. Define the integral of a step function by: ${\displaystyle {\mathsf {I}}\left(\sum _{i=1}^{n}r_{i}{\mathit {1}}_{[x_{i-1},x_{i})}\right)=\sum _{i=1}^{n}r_{i}(x_{i}-x_{i-1})}$. ${\displaystyle {\mathsf {I}}}$ as a function is a bounded linear transformation from ${\displaystyle {\mathcal {S}}}$ into ${\displaystyle \mathbb {R} }$.[1]

Let ${\displaystyle {\mathcal {PC}}}$ denote the space of bounded, piecewise continuous functions on ${\displaystyle [a,b]}$ that are continuous from the right, along with the ${\displaystyle L^{\infty }}$ norm. The space ${\displaystyle {\mathcal {S}}}$ is dense in ${\displaystyle {\mathcal {PC}}}$, so we can apply the B.L.T. theorem to extend the linear transformation ${\displaystyle {\mathsf {I}}}$ to a bounded linear transformation ${\displaystyle {\tilde {\mathsf {I}}}}$ from ${\displaystyle {\mathcal {PC}}}$ to ${\displaystyle \mathbb {R} }$. This defines the Riemann integral of all functions in ${\displaystyle {\mathcal {PC}}}$; for every ${\displaystyle f\in {\mathcal {PC}}}$, ${\displaystyle \int _{a}^{b}f(x)dx={\tilde {\mathsf {I}}}(f)}$.

The Hahn–Banach theorem

The above theorem can be used to extend a bounded linear transformation ${\displaystyle T:S\rightarrow Y}$ to a bounded linear transformation from ${\displaystyle {\bar {S}}=X}$ to ${\displaystyle Y}$, if ${\displaystyle S}$ is dense in ${\displaystyle X}$. If ${\displaystyle S}$ is not dense in ${\displaystyle X}$, 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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