Continuous linear extension

Template:Short description In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space <math>X</math> by first defining a linear transformation <math>L</math> on a dense subset of <math>X</math> and then continuously extending <math>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>L</math> from a normed vector space <math>X</math> to a complete, normed vector space <math>Y</math> can be uniquely extended to a bounded linear transformation <math>\widehat{L}</math> from the completion of <math>X</math> to <math>Y.</math> In addition, the operator norm of <math>L</math> is <math>c</math> if and only if the norm of <math>\widehat{L}</math> is <math>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>[a,b]</math> is a function of the form: <math>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>r_1, \ldots, r_n</math> are real numbers, <math>a = x_0 < x_1 < \ldots < x_{n-1} < x_n = b,</math> and <math>\mathbf{1}_S</math> denotes the indicator function of the set <math>S.</math> The space of all step functions on <math>[a,b],</math> normed by the <math>L^\infty</math> norm (see Lp space), is a normed vector space which we denote by <math>\mathcal{S}.</math> Define the integral of a step function by: <math display=block>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>I</math> as a function is a bounded linear transformation from <math>\mathcal{S}</math> into <math>\R.</math><ref> Here, <math>\R</math> is also a normed vector space; <math>\R</math> is a vector space because it satisfies all of the vector space axioms and is normed by the absolute value function.</ref>

Let <math>\mathcal{PC}</math> denote the space of bounded, piecewise continuous functions on <math>[a,b]</math> that are continuous from the right, along with the <math>L^\infty</math> norm. The space <math>\mathcal{S}</math> is dense in <math>\mathcal{PC},</math> so we can apply the BLT theorem to extend the linear transformation <math>I</math> to a bounded linear transformation <math>\widehat{I}</math> from <math>\mathcal{PC}</math> to <math>\R.</math> This defines the Riemann integral of all functions in <math>\mathcal{PC}</math>; for every <math>f\in \mathcal{PC},</math> <math>\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>T : S \to Y</math> to a bounded linear transformation from <math>\bar{S} = X</math> to <math>Y,</math> if <math>S</math> is dense in <math>X.</math> If <math>S</math> is not dense in <math>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

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References

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