- #1
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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.
(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.