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