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
Sapphire talk enlivens guesswork over iPhone 6
Geneticists offer clues to better rice, tomato crops
UConn makes 3-D copies of antique instrument parts
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: 25,251
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