From the Banach theory we knew that:

1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ and ${\left\| \bullet \right\|_2}$ are two norms defined on $X$, then they are equivalent in the sense that $\exists C_{1},C_{2} \succ 0$ and $C_{1}{\left\| x \right\|_1}\leq {\left\| x \right\|_2} \leq C_{2}{\left\| x \right\|_1}$ for any $x\in X$.

Consider the converse problem now:

2) If any two norms defined on a linear space $X$ are equivalent, then can we be sure that $dimX \prec\infty$?

Moreover, we also know from 1) that any linear space $X$ of finite dimension is complete. Now consider the converse problem:

3) If any norm defined on a linear space $X$ makes the space complete, then can we be sure that $dimX \prec\infty$?

Is this an open problem?

Is there any reference or paper on some sort of problem like this? Or are there any easy counter-example?


Both conditions imply the finite-dimensionality of $X$. If $\dim X = \kappa$ for an infinite cardinal $\kappa$, you can embed $X$ as a dense subspace into $\ell^p(\kappa) = \left\{f \colon\kappa\to\mathbb{K} : \sum_{k\in\kappa} \lvert f(k)\rvert^p < \infty\right\}$ for $1 \leqslant p < \infty$ by mapping each basis element to the "standard basis" element of the same index in $\ell^p(\kappa)$, and for example the induced $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ norms are not equivalent, since the completion in one is reflexive ($\ell^2(\kappa)$ is a Hilbert space), while the completion in the other isn't ($\ell^1(\kappa)^\ast \cong \ell^\infty(\kappa)$, and $\ell^\infty(\kappa)^\ast \not\cong \ell^1(\kappa)$ are shown similarly to the case $\kappa = \mathbb{N}$).

These embeddings also give examples of norms on infinite-dimensional spaces with respect to which the space is not complete.

The norms are naturally given by

$$\lVert f\rVert_p = \left(\sum_{k\in\kappa} \lvert f(k)\rvert^p\right)^{1/p},$$

where the sum over the (possibly) uncountably many non-negative terms $\lvert f(k)\rvert^p$ is defined as

$$\sum_{k\in\kappa} \lvert f(k)\rvert^p := \sup \left\{\sum_{k\in F} \lvert f(k)\rvert^p : F \subset \kappa \text{ finite}\right\}.$$

Of course, if the sum is finite, at most countably many terms of the sum are non-zero.

  • $\begingroup$ learnt a lot, thank you very much. $\endgroup$ – Henry.L Apr 3 '14 at 16:57
  • $\begingroup$ I've added the norms. They are like the $\ell^p(\mathbb{N})$ norms, and for every $f\in \ell^p(\kappa)$ only countably many $f(k)$ can be non-zero. $\endgroup$ – Daniel Fischer Apr 3 '14 at 17:03
  • $\begingroup$ Yes, I came to understand the norm subscript $p$ is meant to be the $l^{p}$. Thanks a lot, I should work harder in the future to avoid such silly problems :-). $\endgroup$ – Henry.L Apr 4 '14 at 0:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.