Continuous linear extension

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

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 $$[a,b]$$ is a function of the form: $$f\equiv r_1 \mathbf{1}_{[a,x_1)}+r_2 \mathbf{1}_{[x_1,x_2)} + \cdots + r_n \mathbf{1}_{[x_{n-1},b]}$$ where $$r_1, \ldots, r_n$$ are real numbers, $$a = x_0 < x_1 < \ldots < x_{n-1} < x_n = b,$$ and $$\mathbf{1}_S$$ denotes the indicator function of the set $$S.$$ The space of all step functions on $$[a,b],$$ normed by the $$L^\infty$$ norm (see Lp space), is a normed vector space which we denote by $$\mathcal{S}.$$ Define the integral of a step function by: $$I \left(\sum_{i=1}^n r_i \mathbf{1}_{ [x_{i-1},x_i)}\right) = \sum_{i=1}^n r_i (x_i-x_{i-1}).$$ $$I$$ as a function is a bounded linear transformation from $$\mathcal{S}$$ into $$\R.$$

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

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