- #1
JamesGold
- 39
- 0
The proof goes as follows:
For contradiction, assume there exists |s|< ∞ such that s = {e1, e2, ... , en} and span(s} = ℝ^∞.
The above makes at least some sense to me. The proof goes on...
Let m > n and u = en+1 + en+2 + ... + em
u [itex]\notin[/itex] span(s), u [itex]\in[/itex] s
Because {e1, e2, ... , en} [itex]\notin[/itex] s, this implies a contradiction. Therefore ℝ^∞ is infinite-dimensional.
I don't see what we just did, and I don't see how what we just did proves that R^infinity is infinite-dimensional. Please help!
For contradiction, assume there exists |s|< ∞ such that s = {e1, e2, ... , en} and span(s} = ℝ^∞.
The above makes at least some sense to me. The proof goes on...
Let m > n and u = en+1 + en+2 + ... + em
u [itex]\notin[/itex] span(s), u [itex]\in[/itex] s
Because {e1, e2, ... , en} [itex]\notin[/itex] s, this implies a contradiction. Therefore ℝ^∞ is infinite-dimensional.
I don't see what we just did, and I don't see how what we just did proves that R^infinity is infinite-dimensional. Please help!