1
$\begingroup$

Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there exists an integer $N$ such that for all $n \geq N$, there exists a non-negative real number $k_n$ for which $p_n \leq k_np_N$.

This is a problem from Dieudonne.

$\endgroup$
2
$\begingroup$

Suppose the condition is satisfied for $N$ and constants $k_n$ for $n \geq N$. Then $p_{N}$ is a norm on $E$ because for every $x \in E$ there is $n$ such that $p_n(x) \gt 0$. If $n \lt N$ then $0 \lt p_n(x) \leq p_N(x)$ and otherwise $0 \lt p_n(x) \leq k_n p_N(x)$. In either case it follows that $p_N(x) \gt 0$. The topology $\tau_N$ on $E$ induced by $p_N$ is clearly coarser than the topology induced by the family $\{p_n\}$. It is also finer because all $p_n$ are continuous with respect to $\tau_N$. Thus, the norm $p_N$ induces the given topology on $E$.

Suppose there is a single norm $\lVert \cdot \rVert$ inducing the topology on $E$. Then all the semi-norms $p_n$ are continuous with respect to the norm $\lVert \cdot \rVert$, so for each $n$ there is $C_n$ such that $p_n(x) \leq C_n \lVert x\rVert$ for all $x \in X$. On the other hand, $\lVert \cdot \rVert$ is continuous with respect to the topology induced by the semi-norms $p_n$, therefore there is $N$ and a constant $C \gt 0$ such that $\lVert x\rVert \leq Cp_{N}(x)$ (since the family $\{p_n\}$ is increasing one semi-norm suffices). We can take $k_n = C_nC$.

$\endgroup$
  • $\begingroup$ where is the local convexity required? $\endgroup$ – Koushik Jan 6 '13 at 13:38
  • 2
    $\begingroup$ Local convexity is a consequence of the fact that the topology can be described by a family of seminorms. Similarly, metrizability would follow from the fact that there is a countable family of semi-norms if we knew in advance that they separate the points (but that would give about half of the problem away). Thus, the hypotheses are a bit redundant. $\endgroup$ – Martin Jan 6 '13 at 13:42
  • $\begingroup$ If "the usual" means that there is a neighborhood basis consisting of balanced, absorbent and convex sets then the Minkowski functionals associated with the elements of the base are a family of semi-norms inducing the topology. (response to a deleted comment which may still be useful) $\endgroup$ – Martin Jan 6 '13 at 13:51

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.