Continuous linear extension

Short description: Mathematical method in functional analysis

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space [math]\displaystyle{ X }[/math] by first defining a linear transformation [math]\displaystyle{ L }[/math] on a dense subset of [math]\displaystyle{ X }[/math] and then continuously extending [math]\displaystyle{ L }[/math] 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 [math]\displaystyle{ L }[/math] from a normed vector space [math]\displaystyle{ X }[/math] to a complete, normed vector space [math]\displaystyle{ Y }[/math] can be uniquely extended to a bounded linear transformation [math]\displaystyle{ \widehat{L} }[/math] from the completion of [math]\displaystyle{ X }[/math] to [math]\displaystyle{ Y. }[/math] In addition, the operator norm of [math]\displaystyle{ L }[/math] is [math]\displaystyle{ c }[/math] if and only if the norm of [math]\displaystyle{ \widehat{L} }[/math] is [math]\displaystyle{ c. }[/math]

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 [math]\displaystyle{ [a,b] }[/math] is a function of the form: [math]\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]} }[/math] where [math]\displaystyle{ r_1, \ldots, r_n }[/math] are real numbers, [math]\displaystyle{ a = x_0 \lt x_1 \lt \ldots \lt x_{n-1} \lt x_n = b, }[/math] and [math]\displaystyle{ \mathbf{1}_S }[/math] denotes the indicator function of the set [math]\displaystyle{ S. }[/math] The space of all step functions on [math]\displaystyle{ [a,b], }[/math] normed by the [math]\displaystyle{ L^\infty }[/math] norm (see Lp space), is a normed vector space which we denote by [math]\displaystyle{ \mathcal{S}. }[/math] Define the integral of a step function by: [math]\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}). }[/math] [math]\displaystyle{ I }[/math] as a function is a bounded linear transformation from [math]\displaystyle{ \mathcal{S} }[/math] into [math]\displaystyle{ \R. }[/math]

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

The Hahn–Banach theorem

The above theorem can be used to extend a bounded linear transformation [math]\displaystyle{ T : S \to Y }[/math] to a bounded linear transformation from [math]\displaystyle{ \bar{S} = X }[/math] to [math]\displaystyle{ Y, }[/math] if [math]\displaystyle{ S }[/math] is dense in [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ S }[/math] is not dense in [math]\displaystyle{ X, }[/math] 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
• Partial function – Function whose actual domain of definition may be smaller than its apparent domain
• Vector-valued Hahn–Banach theorems

References

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