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 T on a dense subset of X and then extending T to the whole space via the theorem below. The resulting extension remains linear and bounded (thus continuous).
This procedure is known as continuous linear extension.
Every bounded linear transformation T from a normed vector space X to a complete, normed vector space Y can be uniquely extended to a bounded linear transformation T ~ from the completion of X to Y . In addition, the operator norm of T is c iff the norm of T ~ is c .
This theorem is sometimes called the B L T theorem, where B L T stands for bounded linear transformation.
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 ≡ r 1 1 [ a , x 1 ) + r 2 1 [ x 1 , x 2 ) + ⋯ + r n 1 [ x n − 1 , b ] where r 1 , … , r n are real numbers, a = x 0 < x 1 < … < x n − 1 < x n = b , and 1 S denotes the indicator function of the set S . The space of all step functions on [ a , b ] , normed by the L ∞ norm (see Lp space), is a normed vector space which we denote by S . Define the integral of a step function by: I ( ∑ i = 1 n r i 1 [ x i − 1 , x i ) ) = ∑ i = 1 n r i ( x i − x i − 1 ) . I as a function is a bounded linear transformation from S into R .
Let P C denote the space of bounded, piecewise continuous functions on [ a , b ] that are continuous from the right, along with the L ∞ norm. The space S is dense in P C , so we can apply the B.L.T. theorem to extend the linear transformation I to a bounded linear transformation I ~ from P C to R . This defines the Riemann integral of all functions in P C ; for every f ∈ P C , ∫ a b f ( x ) d x = I ~ ( f ) .
The above theorem can be used to extend a bounded linear transformation T : S → Y to a bounded linear transformation from 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.
