Discontinuous linear map: Difference between revisions

Revision as of 07:35, 25 August 2022

Revision as of 07:35, 25 August 2022

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== General existence theorem ==

Discontinuous linear maps can be proven to exist more generally even if the space is complete.{{clarify|reason=A general "constructive" proof in the incomplete case was not given above, so this contrast seems kind of hand-waveycomplete.|date=May 2015}} 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''_''n'')_''n'' (<math>n \geq 1</math>) of [[linearly independent]] vectors in ''X''. Define