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

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 ${\displaystyle [a,b]}$ is a function of the form: ${\displaystyle f\equiv r_{1}\mathbf {1} _{[a,x_{1})}+r_{2}\mathbf {1} _{[x_{1},x_{2})}+\cdots +r_{n}\mathbf {1} _{[x_{n-1},b]}}$

where ${\displaystyle r_{1},\ldots ,r_{n}}$ are real numbers, ${\displaystyle a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b,}$ and ${\displaystyle \mathbf {1} _{S}}$ 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:

$${\displaystyle I\left(\sum _{i=1}^{n}r_{i}\mathbf {1} _{[x_{i-1},x_{i})}\right)=\sum _{i=1}^{n}r_{i}(x_{i}-x_{i-1}).}$$

${\displaystyle I}$ 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 BLT theorem to extend the linear transformation ${\displaystyle I}$ to a bounded linear transformation ${\displaystyle {\widehat {I}}}$ 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={\widehat {I}}(f).}$

The Hahn–Banach theorem

The above theorem can be used to extend a bounded linear transformation ${\displaystyle T:S\to Y}$ 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.

See also

• Closed graph theorem (functional analysis)  – Theorems connecting continuity to closure of graphs
• Continuous linear operator
• Densely defined operator  – Function that is defined almost everywhere (mathematics)
• Hahn–Banach theorem  – Theorem on extension of bounded linear functionals
• Linear extension (linear algebra)
• Partial function  – Function whose actual domain of definition may be smaller than its apparent domain
• Vector-valued Hahn–Banach theorems

References

1. Here, ${\displaystyle \mathbb {R} }$ is also a normed vector space; ${\displaystyle \mathbb {R} }$ 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.