Continuous linear extension

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

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