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

From: Stephen J. Herschkorn (herschko_at_rutcor.rutgers.edu)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 06:04:27 GMT

N. Silver wrote:

>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?
>>
>>
>
>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,...}.
>

Wrong. How would you express the vector (1,1,1,...) as a finite linear
combination of elements of this "standard basis"?

By definition, a subset B of a vector space V over a field F iff
for every v in V, there exist *finite* subsets {f1, f2,.., f_n} of
F and {b1, b2,..., b_n} of B such that v = sum (i = 1..n, f_i b_i). 

-- 
Stephen J. Herschkorn                        herschko@rutcor.rutgers.edu

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