Re: a question on orthogonality


David C. Ullrich ha scritto:

On Tue, 16 Dec 2008 01:39:49 -0800 (PST), thx541@xxxxxxxxx wrote:

Hello,
I'm puzzled with the following. Consider the set of polynomials with
real coefficients in one variable, R[x], as a vector space, equipped
with the inner product <\sum_i a_i x^i, \sum_i b_i x^i>= \sum_i
a_i*b_i, which is definite-positive.
Now, take the (infinite) set of polynomials

P= { 1-x, x-x^2, x^2-x^3, ...., x^i-x^(i+1),...}

The sub-space spanned by this set is not R[x] (e.g., 1 and x are not
expressible as l.c. of those polynomials). Now, since <,> above is
definite-positive, I expect that R[x] decomposes as the direct sum of
span(P) and the orthogonal complement of span(P).

Look up the theorem that makes you expect this. That theorem
has a hypothesis that is not satisfied in this example.

However, any nonzero polynomial p(x), of degree say n, is easily
seen not to be orthogonal to x^n-x^(n+1). That is, it seems that the
orthogonal complement of span(P) is just {0}... Clearly, I must be
missing something... Could anyone help?
Thanks

Recalling from memory, the theorem in question says that any subspace
U and its orthogonal complement form a direct sum in a inner product
space (independently of the dimension of this space, which is infinite
in the present case). Now, the only reason that I can figure out for
the theorem not to be applicable to the present case is that the inner
product I defined above is not actually a inner product... But I fail
to see why.


David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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