# Math Help - Infinite dimensional vector Space

1. ## Infinite dimensional vector Space

Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R

2. Originally Posted by math.dj
Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R
For n=1,2,3,..., let $S_n$ be the sequence with a 1 in the n'th position and zeros everywhere else: $S_n = (0,0,\ldots,0,\mathop{1}\limits_{\mathstrut\kern-0.5em {(n\text{'th} \atop \text{coord.})}\kern-0.5em},0,0,\ldots).$

Check that these sequences form an infinite linearly independent set (in the vector space of real sequences with convergent sum).

3. to show that these sequences form a infinite linearly independent set what do i do?

4. Originally Posted by math.dj
to show that these sequences form a infinite linearly independent set what do i do?
For a given natural number N, show that the set $\{S_1,S_2,\ldots,S_N\}$ is linearly independent. That shows that the dimension of the space is at least N. Since that holds for all N, it follows that the space is infinite-dimensional.

5. Thank you for your help ...U Rock