Span of a linearly independent subset of a hilbert space is a subspace iff finite

[waddles]

Homework Statement

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.

Homework Equations

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.

 

[sunjin09]

Sorry I have no idea

 

[Dick]

Homework Statement

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.

Homework Equations

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.

span(S) is the set of all FINITE linear combinations of elements in S. To see how that would make a problem for the set being closed if S is infinite, define a convergent series that contains multiples of an infinite number of elements of S.