Exercise 37, § 2

a) Let A be an algebra over $\mathbf{C}$ endowed with a locally convex topology. The following conditions are equivalent:

(i) there exists a fundamental system $(U_i)$ of convex balanced neighbourhoods of 0 such that $U_iU_i\subset U_i$ for every $i$;

(ii) A is isomorphic to a subalgebra of a product of normed algebras;

(iii) the topology of A can be defined by a family of semi-norms $p_i$ satisfying $p_i(xy)\leqslant p_i(x)p_i(y)$ for all $x, y\in A$.

If A satisfies these conditions, A is said to be locally $m$-convex.

b) Let A be a locally $m$-convex algebra. The mapping $(x, y)\mapsto xy$ from $A\times A$ into A is continuous. If A is unital, and if G denotes the set of invertible elements of A, then the mapping $x\mapsto x^{-1}$ from G into A is continuous.

c) Let A be a unital locally $m$-convex algebra. The spectrum of every element of A is nonempty. If A is a field, then $A =\mathbf{C}\cdot 1$.

d) Let A be a complete locally $m$-convex algebra. There exist an increasing ordered filtered set I, a family $(A_i)_{i\in I}$ of Banach algebras, continuous morphisms $\pi_{ij}: A_j\mapsto A_i$ for $i\leqslant j$, with $\pi_{ij}\circ \pi_{jk}=\pi_{ik}$, such that A is isomorphic to the subalgebra of $\prod_iA_i$ formed by the $(x_i)$ such that $\pi_{ij}(x_j) =x_i$ for $i\leqslant j$.

e) The algebra $\mathscr{C}_{\mathbf{C}}(\mathbf{R})$, endowed with the topology of compact convergence, is locally $m$-convex. The set of invertible elements of $\mathscr{C}_{\mathbf{C}}(\mathbf{R})$ is dense in $\mathscr{C}_{\mathbf{C}}(\mathbf{R})$; it is not open.

f) Let D be an open subset of $\mathbf{C}$ and A the algebra of complex holomorphic functions in D, endowed with the topology of compact convergence. Then A is locally $m$-convex. The set of invertible elements of A is closed.

g) The algebra of indefinitely differentiable complex functions on $[0,1]$, endowed with the topology of uniform convergence of each derivative, is locally $m$-convex.

h) Let A be the algebra of (classes of) complex functions $f$ on $[0,1]$ such that $f\in L^p([0,1])$ for every $p >1$. Endowed with the topology defined by the family of semi-norms $f\mapsto \|f\|_p(p >1)$, A is not locally $m$-convex. The mapping $(x, y)\mapsto xy$ from $A\times A$ into A is continuous.