11.4 Barreled Spaces
Definition 11.4.1 (Barrel).label Let $E$ be a TVS over $K \in \RC$ and $D \subset E$, then $D$ is a barrel if it is convex, circled, radial, and closed.
Definition 11.4.2 (Barreled Space).label Let $E$ be a locally convex space over $K \in \RC$, then the following are equivalent:
- (1)
The barrels of $E$ forms a fundamental system of neighbourhoods at $0$.
- (2)
Every barrel in $E$ is a neighbourhood of $0$.
- (3)
Every lower semicontinuous seminorm on $E$ is continuous.
Proof. $(2) \Rightarrow (1)$: Let $\fB \subset \cn_{E}(0)$ be a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets, then $\ol{\fB}= \bracsn{\ol U|U \in \fB}$ is a fundamental system of neighbourhoods at $0$ consisting of barrels.
$(2) \Rightarrow (3)$: Let $\rho: E \to [0, \infty)$ be a lower semicontinuous seminorm, then $\bracs{\rho > 1}$ is open and $\bracs{\rho \le 1}$ is a Barrel. In which case, $\rho$ is continuous by (4) of Lemma 11.1.9.
$(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of Definition 11.1.11, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_{E}(0)$ by (5) of Lemma 11.1.9.$\square$
Summary 11.4.3.label The following types of locally convex spaces are barrelled:
- (1)
Every locally convex space with the Baire property.
- (2)
Every Banach space and every Fréchet space.
- (3)
Inductive limits of barrelled spaces.
- (4)
Spaces of type (LB) and (LF).
- (5)
The locally convex direct sum of barrelled spaces.
- (6)
Products of barrelled spaces.
Proposition 11.4.4.label Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barrelled.
Proof, [II.7.1, SW99]. Let $D \subset E$ be a Barrel, then $E = \bigcup_{n \in \natp}nD$ is a countable union of closed sets. Since $E$ is Baire, there exists $n \in \natp$, $U \in \cn_{E}(0)$ circled, and $x \in E$ such that $x + U \in nB$. In which case,
so $2nB$ and thus $B$ is a neighbourhood of $0$.$\square$
Proposition 11.4.5.label Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_{i} \in \hom(E_{i}; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barrelled.
Proof, [II.7.2, SW99]. Let $D \subset E$ be a barrel, then for each $i \in I$, $T_{i}^{-1}(D) \subset E_{i}$ is also a barrel, and thus a neighbourhood of $0$ in $E_{i}$. By (5) of Definition 11.8.1, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barrelled.$\square$
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