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Line 45: == General existence theorem == Discontinuous linear maps can be proven to exist more generally, even if the space is ~~not~~ complete. Let ''X'' and ''Y'' be [[normed space]]s over the field ''K'' where <math>K = \R</math> or <math>K = \Complex.</math> Assume that ''X'' is infinite-dimensional and ''Y'' is not the zero space. We will find a discontinuous linear map ''f'' from ''X'' to ''K'', which will imply the existence of a discontinuous linear map ''g'' from ''X'' to ''Y'' given by the formula <math>g(x) = f(x) y_0</math> where <math>y_0</math> is an arbitrary nonzero vector in ''Y''. If ''X'' is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing ''f'' which is not bounded. For that, consider a [[sequence]] (''e''<sub>''n''</sub>)<sub>''n''</sub> (<math>n \geq 1</math>) of [[linearly independent]] vectors in ''X''. Define

Discontinuous linear map: Difference between revisions