W^3
Posts:
28
Registered:
4/19/11
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Re: Dimension of the space of real sequences
Posted:
Nov 14, 2012 9:19 PM
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If R^N had a countable basis, then so would every subspace of R^N. In
particular l^2 would have a countable basis, call it {v_1,_2, ...}.
Setting V_n = span {v_1, ..., v_n}, we then have l^2 = V_1 U V_2 U ...
But this violates Baire, as l^2 is complete (in its usual metric) and
each V_n is closed and nowhere dense in l^2.
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