Register to reply

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

Share this thread:
waddles
#1
Mar5-12, 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.
Phys.Org News Partner Science news on Phys.org
Scientists discover RNA modifications in some unexpected places
Scientists discover tropical tree microbiome in Panama
'Squid skin' metamaterials project yields vivid color display
sunjin09
#2
Mar5-12, 01:21 PM
P: 312
Sorry I have no idea
Dick
#3
Mar5-12, 04:13 PM
Sci Advisor
HW Helper
Thanks
P: 25,235
Quote Quote by waddles View Post
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.
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.


Register to reply

Related Discussions
Find a linearly independent subset F of E Calculus & Beyond Homework 2
The dimension of the span of three linearly independent R^3 vectors Calculus & Beyond Homework 6
Linearly independent vectors and span Calculus & Beyond Homework 3
Lin. Alg. - Is the set a linearly independent subset of R^3 Calculus & Beyond Homework 5
Subset linearly independent? Introductory Physics Homework 9