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This is cross-posted and answered on MO here.

Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-complete if every P-sequence converges. Problem 1133 of the College Mathematics Journal (proposed by Kirk Madsen, solved by Eugene Herman) asks you to prove that $$\text{compact}\Longrightarrow\text{P-complete}\Longrightarrow\text{complete}$$ and that none of these implications go both ways. The implications follow by showing that $$\text{sequence}\Longleftarrow\text{P-sequence}\Longleftarrow\text{Cauchy sequence},$$ since a P-sequence (and thus a Cauchy sequence) converges iff it has a convergent subsequence. To give counterexamples to the converses, there are several possible directions. My question specifically involves normed vector spaces (although it is overkill for the original problem).

For any $n\geq 0$, any norm on $\mathbb R^n$ induces a P-complete metric. This distinguishes compactness and P-completeness, since $\mathbb R^n$ obviously isn't compact when $n>0$. To differentiate P-completeness and completeness, we can note that a Hilbert space is P-complete iff it is finite-dimensional (otherwise, we take a non-repeating sequence of vectors from an orthonormal basis and get a P-sequence that doesn't converge). I wonder if other infinite-dimensional normed spaces (necessarily Banach) might be P-complete. But my knowledge of Banach spaces is very limited, so I don't have much intuition about what examples to try. Also, the property of P-completeness (unlike compactness and completeness) is not closed-hereditary, so we can't just try an something by embedding it in a larger example.

Question: What is an example of an infinite dimensional, P-complete Banach space?

Examples I tried:

  • $\ell^p$ spaces for all $1\leq p<\infty$. They are not P-complete, since the sequence $e_n=(0,\dots,0,1,0,\dots)$ is a P-sequence but not Cauchy.
  • $C(X)$ for $X$ compact Hausdorff, first-countable and infinite. There must be an accumulation point $p\in X$. We can take a sequence of bump functions $f_k$ converging (pointwise) to the characteristic function $\chi_p$. For any $g\in C(X)$, we have $\lim d(g,f_k)=||g-\chi_p{||}_\infty$. Thus $(f_k)$ is a P-sequence that does not converge (uniformly), because the pointwise limit is discontinuous.
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One idea is to try and modify your current construction. You've noted that $\ell^p$ is not P-complete by considering the sequence $e_{n}$. Let's take $\ell^{1}$, for simplicity, and adjust the metric a little bit, so that the distance between two sequences $a_{n}$ and $b_{n}$ is $\sum_{n}\frac{1}{n^2}|a_n - b_n|$. Using this metric (and the corresponding norm):

  1. Do we obtain a Banach space?
  2. Does $e_n$ still provide a counterexample?
  3. Is our space P-complete?

I haven't worked out the answers to these questions, but this might be an interesting direction to think about.

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  • $\begingroup$ I think $n^2e_n$ is a non-convergent P-sequence. It converges to $0$ component-wise, but each element has norm $1$, so it can't be convergent in this norm. For any sequence $a_n$ with finite norm, we have $$\lim_{k\rightarrow\infty}\sum_n\frac{1}{n^2}|a_n-n^2\delta_{nk}|=\lim_{k\rightarrow\infty}\sum_n|a_n/n^2-\delta_{nk}|=1+\sum_n|a_n/n^2|,$$ since we must have $a_n/n^2\rightarrow 0$ to have finite norm. Thus $n^2e_n$ is a P-sequence. While I haven't worked out the details, this makes me think that modifying any $\ell^p$ by scaling by a sequence of non-zero real numbers won't work. $\endgroup$ – Nikhil Sahoo Jul 22 at 21:12
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Disclaimer: I am providing an answer so that the question will appear on MSE as answered. This is not my answer, but rather the work of Bill Johnson and Mikhail Ostrovskii. All credit to them. For details, see the MO cross-post.

Infinite-dimensional, P-complete Banach spaces are plentiful. In fact, every Banach space is a subspace of a P-complete Banach space (Bill Johnson proves this in the accepted answer to the MO cross-post). The examples constructed are quite large, using transfinite induction in two separate stages (transfinite induction is used to embed a space $X$ into a larger space $Z$; this process is then iteratedly transfinitely, a total of $\omega_1$ many times). To see that "large" examples are necessary, we can look at the answer of Mikhail Ostrovskii (which is not the accepted answer, but it still great!), which proves that an infinite-dimensional, P-complete Banach space cannot be separable.

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