In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space on a dense subset and then continuously extending 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

from the completion In addition, the operator norm if and only if

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 denotes the indicator function norm (see Lp space

denote the space of bounded, piecewise

The Hahn–Banach theorem

if then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.

See also

References