Exact definition of boundedness in a Frechet space : bounded with respect to “each” seminorm

functional-analysisgeneral-topology

Let $V$ be a Frechet space with the seminorms given by $\lVert \cdot \rVert_n$ for $n=1,2,3, \cdots$.

Then, in the Wikipedia https://en.wikipedia.org/wiki/Bounded_set_(topological_vector_space), it is stated that a set $B \subset V$ is bounded if it is bounded with respect to all $\lVert \cdot \rVert_n$.

Does it mean

  1. $\lVert x \rVert_n \leq C_n$ for all $x \in B$ and each $n$ or
  2. $\lVert x \rVert_n \leq C$ for all $x \in B$ and all $n$?

where $C_n$'s and $C$ are some positive constants. The former depends on $n$ while the latter doesn't.

Wikipedia is somewhat obscure in stating the property, so I am confused. Certainly, the second condition is much stronger than the first condition, but which one exactly corresponds to boundedness in a Frechet space?

Could anyone please clarify?

Best Answer

One can define boundedness for a general topological vector space $(X, \tau)$ in a more geometric way (independently of any seminorm):

A set $S \subset X$ is bounded if for every neighborhood $V$ of the origin there is a $\lambda> 0$ such that $S \subset \lambda V$.

Now, it is easy to show that if $\tau$ is a locally convex topology, induced by the family of seminorms $\{\lVert \cdot \rVert_i\}_{i \in \mathbb{N}}$, then the above is equivalent to requiring that, for every $i \in \mathbb{N}$, $\lVert S \rVert_i = \{\lVert x \rVert_i \; : \; x \in S\}$ is bounded (in $\mathbb{R}$). That's condition (1) in your question.

Related Question