# Non-convergence counter example?

1. Jul 27, 2010

### strangerep

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 $\Xi$ of arbitrary-length
sequences over C (the complex numbers). Denote

$$\delta_k ~:=~ (0,0,\dots,0,1,0,0,\dots) ~~, ~~~~~ \mbox{where the 1 is in the k-th position.}$$

Every $f\in\Xi$ can be expressed as
$$f ~:=~ \sum_{k=0}^\infty \, f_k \, \delta_k ~~~~~ \mbox{where all}~ f_k \in C.$$

For arbitrary $f,g \in \Xi$, we denote the usual positive definite
(left antilinear) Hermitian formal inner product $(\cdot,\cdot)$ by

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

Such an inner product is ill-defined (divergent) in general, and one
usually imposes restrictions on the allowed coefficients $f_k$ to
ensure the inner product is well-defined. For example, one might demand
that $f_k=0$ for all $k$ greater than some integer $N_f$ (different in
general for each $f$). Denote this space as V. Alternatively, one could
demand that the $\sum_k|f_k|^2$ 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., $H^* = H$.

But my question concerns the universal vector space $\Xi$ above,
in which there are no restrictions on the coefficients in the sequences,
and this expression:

$$P(g,f) ~:=~ {\frac{|(g,f)|^2}{(g,g) \, (f,f)}} ~~, ~~~~~~~~~(f,g \ne 0) ~.$$

Since the numerator and/or either factor in the denominator may be
divergent, I'm interpreting this expression in the following sense:

$$P(g,f) ~:=~ \lim_{N\to\infty} \, P_N(g,f) \, ~:=~ \lim_{N\to\infty} \, \frac{|(g,f)_N|^2} {\, (g,g)_N \; (f,f)_N} ~~, ~~~~~~~~~(f,g \ne 0) ~,$$

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 $P_N(g,f)$
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
$P_N(g,f)$ 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.