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In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space X {\displaystyle X} by first defining a linear transformation T {\displaystyle {\mathsf {T}}} on a dense subset of X {\displaystyle X} and then extending T {\displaystyle {\mathsf {T}}} to the whole space via the theorem below. The resulting extension remains linear and bounded.

This procedure is known as continuous linear extension.