Editing Lp space (section)

===Closed subspaces===

If <math>0 < p < \infty</math> is any positive real number, <math>\mu</math> is a [[probability measure]] on a measurable space <math>(S, \Sigma)</math> (so that <math>L^\infty(\mu) \subseteq L^p(\mu)</math>), and <math>V \subseteq L^\infty(\mu)</math> is a vector subspace, then <math>V</math> is a closed subspace of <math>L^p(\mu)</math> if and only if <math>V</math> is finite-dimensional{{sfn|Rudin|1991|pp=117–119}} (<math>V</math> was chosen independent of <math>p</math>). 
In this theorem, which is due to [[Alexander Grothendieck]],{{sfn|Rudin|1991|pp=117–119}} it is crucial that the vector space <math>V</math> be a subset of <math>L^\infty</math> since it is possible to construct an infinite-dimensional closed vector subspace of <math>L^1\left(S^1, \tfrac{1}{2\pi}\lambda\right)</math> (which is even a subset of <math>L^4</math>), where <math>\lambda</math> is [[Lebesgue measure]] on the [[unit circle]] <math>S^1</math> and <math>\tfrac{1}{2\pi} \lambda</math> is the probability measure that results from dividing it by its mass <math>\lambda(S^1) = 2 \pi.</math>{{sfn|Rudin|1991|pp=117–119}}