The discussion focuses on proving that the span of a linearly independent subset S of a Hilbert space is a subspace, specifically a linear manifold and a closed set, if and only if S is finite. It is established that if S is finite, then span(S) is closed due to the properties of finite subsets in metric spaces. Conversely, if S is infinite, span(S) may fail to be closed, as demonstrated by constructing a convergent series involving infinitely many elements of S.
PREREQUISITESMathematics students, particularly those studying functional analysis, linear algebra, and anyone interested in the properties of Hilbert spaces and their subspaces.
waddles said: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.