(adsbygoogle = window.adsbygoogle || []).push({}); Non-convergence counter example??

(This question occurred to me in the context of quantum field theory,

but since it's purely mathematical, I'm asking it here...)

Consider the universal vector space [itex]\Xi[/itex] of arbitrary-length

sequences over C (the complex numbers). Denote

[tex]

\delta_k ~:=~ (0,0,\dots,0,1,0,0,\dots) ~~,

~~~~~ \mbox{where the 1 is in the k-th position.}

[/tex]

Every [itex]f\in\Xi[/itex] can be expressed as

[tex]

f ~:=~ \sum_{k=0}^\infty \, f_k \, \delta_k

~~~~~ \mbox{where all}~ f_k \in C.

[/tex]

For arbitrary [itex]f,g \in \Xi[/itex], we denote the usual positive definite

(left antilinear) Hermitian formal inner product [itex](\cdot,\cdot)[/itex] by

[tex]

(f,g) ~:=~ \sum_{k=0}^\infty\, \bar{f_k} \, g_k ~~.

[/tex]

Such an inner product is ill-defined (divergent) in general, and one

usually imposes restrictions on the allowed coefficients [itex]f_k[/itex] to

ensure the inner product is well-defined. For example, one might demand

that [itex]f_k=0[/itex] for all [itex]k[/itex] greater than some integer [itex]N_f[/itex] (different in

general for each [itex]f[/itex]). Denote this space as V. Alternatively, one could

demand that the [itex]\sum_k|f_k|^2[/itex] is finite, this space being denoted by H

and is Cauchy-complete in the norm topology induced by the inner product.

It's well known that H is self-dual, i.e., [itex]H^* = H[/itex].

But my question concerns the universal vector space [itex]\Xi[/itex] above,

in which there are no restrictions on the coefficients in the sequences,

and this expression:

[tex]

P(g,f) ~:=~ {\frac{|(g,f)|^2}{(g,g) \, (f,f)}}

~~, ~~~~~~~~~(f,g \ne 0) ~.

[/tex]

Since the numerator and/or either factor in the denominator may be

divergent, I'm interpreting this expression in the following sense:

[tex]

P(g,f)

~:=~ \lim_{N\to\infty} \, P_N(g,f) \,

~:=~ \lim_{N\to\infty} \, \frac{|(g,f)_N|^2}

{\, (g,g)_N \; (f,f)_N} ~~,

~~~~~~~~~(f,g \ne 0) ~,

[/tex]

where the "N" subscripts mean that we have truncated the vectors

to N terms (i.e., restricted to an N-dimensional subspace).

By the usual Cauchy-Schwarz inequality, P(g,f) is bounded in [0,1],

since every term of the sequence is so-bounded (i.e., for any value

of N). This is true even if the numerator or denominator diverge

separately as N increases arbitrarily (afaict)

But here's my question: does the sequence necessarily converge?

Or are there examples of f,g such that [itex]P_N(g,f)[/itex]

wanders back and forth inside the range [0,1] forever with

increasing N, never converging to a particular value therein?

The random examples I've tried (where f and/or g have divergent

norm, and/or (f,g) is divergent) all seem to converge to either

0 or 1. I can't find any examples that oscillate back and

forth forever in a non-convergent manner.

Can anyone out there think of an example of f,g such that the sequence of

[itex]P_N(g,f)[/itex] oscillates forever with increasing N? Or am I overlooking

some standard result which says that such sequences do indeed always converge??

Thanks in advance for any help/suggestions.

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# Non-convergence counter example?

Can you offer guidance or do you also need help?

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