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Span of a linearly independent subset of a hilbert space is a subspace iff finite 
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#1
Mar512, 02:27 AM

P: 4

1. The problem statement, all variables and given/known data
Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite. 2. Relevant equations 3. The attempt at a solution Assuming S is finite means that S is a closed set (a finite subset of a metric space is closed). I think that this will help to prove span(S) is a closed set but I am a bit stuck. 


#2
Mar512, 01:21 PM

P: 312

Sorry I have no idea



#3
Mar512, 04:13 PM

Sci Advisor
HW Helper
Thanks
P: 25,228




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