Exercise 26, § 5

Let $E$ be an infinite dimensional Fréchet space of enumerable type. a) Show that there exists a sequence $(a_n)$ of linearly independent elements of $E$ such that each sequence $(a_{2n})$ and $(a_{2n+1})$ is total (use exerc. 24, c). b) Let $F$ be the vector subspace of $E$ generated by the $a_{2n+1}$ ($n \in \mathbf{N}$). For every $n > 0$ let $M_n$ be the subspace generated by the $a_{2k}$ with $k \leq n$. For each $n$, let $\phi_n$ be the restriction to $F$ of the canonical homomorphism of $E$ on $E/M_n$, and let $\mathscr{T}_n$ be the topology on $F$ which is the inverse image under $\phi_n$ of the quotient topology, on $E/M_n$. Show that each of the topologies $\mathscr{T}_n$ on $F$ is a Hausdorff locally convex topology, but that the lower bound of the $\mathscr{T}_n$ in the set of locally convex topologies on $F$ is the coarsest topology on $F$. \* c) Take $E$ to be a Hilbert space; show that we can choose the sequence $(a_n)$ so that if $G$ is the closed vector subspace generated by the $a_{4n+1}$, then $G$ has infinite codimension and so that the images of the $a_{2n}$ and the $a_{4n+3}$ in $E/G$ are still linearly independent. Write $G_n$ for the subspace of $E$ that is the sum of $G$ and of the subspace generated by the $a_{4k+3}$ with $k \leq n$, and give to $G_n$ the topology which is the inverse image under the canonical mapping restricted to $G_n$, of the quotient topology on $E/M_n$. Show that the sequence $(G_n)$ is an inductive system of topological vector spaces such that $G_n$ is closed in $G_{n+1}$ for the topology of $G_{n+1}$, but that $G_n$ is not closed in the inductive limit space of this sequence. \*