Articles of topological vector spaces

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology on a vector space is it enough to show that: $P$ is a family of semi-norms on $V$ $P$ if separating […]

About functions of bounded variation

I got the following the following idea in one of the articles that I’m reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions of the compact interval $[a,b]$. For function $g:[a,b]\to X$ and each $D\in \mathcal{D}$, we write $$V(g,D)=(D)\sum [g(v)-g(u)]$$ where $$D=\{[u,v]\}$$ […]

When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the topology of uniform convergence on bounded subsets on $E$). Under what assumptions on $E$ is it possible to conclude that […]

Openness of linear mapping

Let $X$ be a topological vector space over the field $K$, where $K=\mathbb R$ or $K= \mathbb C$, and let $f:X\rightarrow K^n$ ($n \in \mathbb N$) be a linear and surjective functional. How to prove that $f$ is an open mapping? Thanks

Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that there exists a local base of absolutely convex absorbent sets is crucial. Thanks for suggestions or tips.

If $V$ is completely normable, then is every norm complete?

Here is a theorem that motivated my question. Let $(V,||\cdot||_V)$ be a normed space over $\mathbb{K}$. Then, there exists a Banach space $(X,||\cdot||_X)$ such that $V$ is dense in $X$ and $||\cdot||_X$ is an extension of $||\cdot||_V$. Let $(V,\tau)$ be a normable space over $\mathbb{K}$. Assume that there exists a norm $||\cdot||$ on $V$ such […]

Closedness in the proof of the Alaoglu Theorem

I’m reading a proof the Bourbaki-Alaoglu Theorem: Could someone explain how the closedness of $\Phi(V^\circ)$ (namely, $\Phi(V^\circ)$ contains all of its limit points) is done in the proof? I totally don’t understand the step of “continuity of the coordinate functions shows that …”. (Where is the continuity used?)

Different definitions of absorbing sets from the Wikepedia

Consider a vector space $X$ over the field $\mathbb{F}$ of real or complex numbers and a set $S\subset X$. In this Wikipedia article about absorbing sets, $S$ is called absorbing if for all $x\in X$ there exists a real number $r$ such that for all $\alpha\in\mathbb{F}$ with $\vert \alpha \vert \geq r$ we have $$ […]

Given a Banach space $X$, are weak$^*$ bounded subsets of the dual space $X '$ also strongly bounded (with respect to the usual norm in $X '$)?

Some related facts I already know: 1) In a Banach space $X$, weakly bounded sets are strongly bounded and vice-versa (Thm 3.18 – “Functional Analysis“, Rudin); 2) From 1, it follows that my question is equivalent to proving that weak$^*$ bounded sets are bounded with respect to the weak topology of the dual $X’$. 3) […]

How to prove in a topological vector space: cl(A) + cl(B) is a subset of cl(A+B), where cl denotes closure?

I’m not sure where to really proceed. My process is as follows. Take any $x \in cl(A)+cl(B)$. Assume for a contradiction that $x \notin cl(A+B)$. Then there exists an open set $U$ such that $ x \in U$, and $U \cap (A+B) = \emptyset$. So for all $u \in U$ and $b \in B$ we […]