Date: Nov 14, 2012 2:24 AM Author: Jose Carlos Santos Subject: Re: Dimension of the space of real sequences On 14-11-2012 5:21, David Bernier wrote:

>> Can someone please tell me how to prove that the real vector space of

>> all sequences of real numbers has uncountable dimension?

>

> Suppose the n't term of some sequence is s_n.

>

> Then the set S of vectors 's' such that s_n = O(1/n^2)

> is also vector space ...

>

>

> Hint:

> s_n = O(1/n^2) means "Exists K, a real number, |s_n| <= K * (1/n^2) " .

>

> So S is a real vector space.

> All elements of S are elements of the real Hilbert space l^2

> of square-summable sequences of reals .

>

> S is vector subspace of l^2, the real infinite-dimensional Hilbert

> space. [ infinite dimensional: with Hilbert spaces, one normally

> classifies spaces by the cardinality of a Hilbert basis of

> the Hilbert space].

>

>

> By what appears below, a Hamel basis (vector space basis)

> of the infinite dimensional

> real vector space l^2 does not have a countable basis.

> So S does not have a countable basis.

>

>

> Background at PlanetMath

> =========================

>

> There's a Panet Math entry with title:

> "Banach spaces of infinite dimension do not have a countable Hamel basis" :

> http://planetmath.org/encyclopedia/ABanachSpaceOfInfiniteDimensionDoesntHaveACountableAlgebraicBasis.html

>

>

> Usually, when nothing is said, statements on Banach spaces

> apply irrespective of whether it is a real Banach space

> or a complex one ...

>

> The proof at PlanetMath appeals to the Baire Category Theorem,

> http://en.wikipedia.org/wiki/Baire_category_theorem

>

> "(BCT1) Every complete metric space is a Baire space."

>

> As I recall, when using the form I know of BCT, we always want

> the metric space to be complete.

>

>

> Banach spaces are complete normed spaces .

>

>

>

> They cite a Monthly article:

> 1 H. Elton Lacey, The Hamel Dimension of any Infinite Dimensional

> Separable Banach Space is c, Amer. Math. Mon. 80 (1973), 298.

>

> ( c = cardinality of the continuum ).

>

> A Hamel basis is a basis in the sense of vector spaces (only finite sums).

>

> A Hilbert basis for a Hilbert space :

> The Wikipedia article on Orthonormal basis says this about

> "Hilbert basis":

>

> "Note that an orthonormal basis in this sense is not generally a Hamel

> basis, since infinite linear combinations are required."

>

> cf.:

>

> http://en.wikipedia.org/wiki/Orthonormal_basis

>

>

> Baire space: [at Wikipedia] Any countable intersection of dense open

> sets, is itself dense.

>

> N.B.: There's a way of switching things around by looking

> at the complements, which are then closed sets.

> [ By De Morgan's laws, if I'm not mistaken ].

Thanks a lot.

Best regards,

Jose Carlos Santos