Continuous linear extension

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space ${\displaystyle X}${\displaystyle X} by first defining a linear transformation ${\displaystyle L}${\displaystyle L} on a dense subset of ${\displaystyle X}${\displaystyle X} and then continuously extending ${\displaystyle L}${\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.

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

This theorem is sometimes called the BLT theorem.

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval ${\displaystyle [a,b]}${\displaystyle [a,b]} is a function of the form: ${\displaystyle f\equiv r_{1}\mathbf {1} _{[a,x_{1})}+r_{2}\mathbf {1} _{[x_{1},x_{2})}+\cdots +r_{n}\mathbf {1} _{[x_{n-1},b]}}${\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 ${\displaystyle r_{1},\ldots ,r_{n}}${\displaystyle r_{1},\ldots ,r_{n}} are real numbers, ${\displaystyle a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b,}${\displaystyle a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b,} and ${\displaystyle \mathbf {1} _{S}}${\displaystyle \mathbf {1} _{S}} denotes the indicator function of the set ${\displaystyle S.}${\displaystyle S.} The space of all step functions on ${\displaystyle [a,b],}${\displaystyle [a,b],} normed by the ${\displaystyle L^{\infty }}${\displaystyle L^{\infty }} norm (see Lp space), is a normed vector space which we denote by ${\displaystyle {\mathcal {S}}.}${\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\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}).} ${\displaystyle I}${\displaystyle I} as a function is a bounded linear transformation from ${\displaystyle {\mathcal {S}}}${\displaystyle {\mathcal {S}}} into ${\displaystyle \mathbb {R} .}${\displaystyle \mathbb {R} .}^[1]

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

The above theorem can be used to extend a bounded linear transformation ${\displaystyle T:S\to Y}${\displaystyle T:S\to Y} to a bounded linear transformation from ${\displaystyle {\bar {S}}=X}${\displaystyle {\bar {S}}=X} to ${\displaystyle Y,}${\displaystyle Y,} if ${\displaystyle S}${\displaystyle S} is dense in ${\displaystyle X.}${\displaystyle X.} If ${\displaystyle S}${\displaystyle S} is not dense in ${\displaystyle X,}${\displaystyle X,} then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.

• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Continuous linear operator – Function between topological vector spaces
• Densely defined operator – Linear operator on dense subset of its apparent domain
• Hahn–Banach theorem – Theorem on extension of bounded linear functionals
• Linear extension (linear algebra) – Mathematical function, in linear algebraPages displaying short descriptions of redirect targets
• Partial function – Function whose actual domain of definition may be smaller than its apparent domain
• Vector-valued Hahn–Banach theorems

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

• Functional analysis
• Linear operators

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