Continuous linear extension
In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space formula_1 by first defining a linear transformation formula_2 on a dense subset of formula_1 and then extending formula_2 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. Theorem. Every bounded linear transformation formula_2 from a normed vector space formula_1 to a complete, normed vector space formula_7 can be uniquely extended to a bounded linear transformation formula_8 from the completion of formula_1 to formula_10  In addition, the operator norm of formula_2 is formula_12 if and only if the norm of formula_8 is formula_14 This theorem is sometimes called the B L T theorem, for "bounded linear transformation". Application. Consider, for instance, the definition of the Riemann integral. A step function on a closed interval formula_15 is a function of the form: formula_16 where formula_17 are real numbers, formula_18 and formula_19 denotes the indicator function of the set formula_20 The space of all step functions on formula_21 normed by the formula_22 norm (see Lp space), is a normed vector space which we denote by formula_23  Define the integral of a step function by: formula_24 formula_25 as a function is a bounded linear transformation from formula_26 into formula_27 Let formula_28 denote the space of bounded, piecewise continuous functions on formula_15 that are continuous from the right, along with the formula_22 norm. The space formula_26 is dense in formula_32 so we can apply the BLT theorem to extend the linear transformation formula_25 to a bounded linear transformation formula_34 from formula_28 to formula_27 This defines the Riemann integral of all functions in formula_28; for every formula_38 formula_39 The Hahn–Banach theorem. The above theorem can be used to  extend a bounded linear transformation formula_40 to a bounded linear transformation from formula_41 to formula_42 "if" formula_43 is dense in formula_44 If formula_43 is not dense in formula_46 then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.