Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?

From: N. Silver (mathelp_at_worldnet.att.net)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 03:07:51 GMT

Ron Jones wrote:

> I'd be very grateful if anyone could answer the following.
> If V is the set of all countable-tuples (a_1,a_2,a_3,...)
> where a_i are rational or real, then under usual pointwise
> addition and scalar multiplication it is a vector space.
> Every vector space has a basis - what then is a basis of V?
> Also any book that deals with infinite-dimensional spaces-...

The standard basis is:
e1 = (1,0,0,...),
e2 = (0,1,0,...),
e3 = (0,0,1,...),...

Then a vector (a_1,a_2,a_3,...) = (a_1)e1 + (a_2)e2 +...

This space is isomorphic to the space of polynomials,
however you want to choose the coefficients (either only
rationals or real coefficients). The standard basis for this
space is {1, x, x^2, x^3,...}.

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