Appendix

Continuous polynomials and formal series

In this Appendix, F denotes a separated polynormed space over K, $E_i$ (for $1 \leq i \leq n$) a normed space over K and E the topological vector space product of the $E_i$.

For $\alpha \in \mathbf{N}^n$, and $1 \leq j \leq |\alpha|$, we put:

(the sequence of $\alpha(j)$ is therefore obtained by writing $\alpha_1$ times 1, ..., $\alpha_n$ times n).

We denote by $E_\alpha$ the topological vector space product of the family of the $E_{\alpha(j)}$ for $1 \leq j \leq |\alpha|$. We denote by $\mathrm{Hom}_\alpha(E_1, \ldots, E_n; F)$ the space of $|\alpha|$-multilinear mappings from $E_\alpha$ into F and by $\mathscr{L}_\alpha(E_1, \ldots, E_n; F)$ the subspace of $\mathrm{Hom}_\alpha(E_1, \ldots, E_n; F)$ formed by the continuous multilinear mappings endowed with the topology of uniform convergence on the bounded subsets of $E_\alpha$; it is a separated polynormed space, whose topology can be defined by the family of semi-norms $\|u\|_y$, where, for a continuous semi-norm $y$ on F, we denote by $\|u\|_y$ the greatest lower bound of the numbers $a \geq 0$ such that

where each $x_j$ runs through $E_{\alpha(j)}$.

We denote by $p_i$ the canonical projection of E onto $E_i$. For $\alpha \in \mathbf{N}^n$ with $\alpha \neq 0$, we denote by $p_\alpha$ the mapping $x \mapsto (p_{\alpha(j)}(x))$ from E into $E_\alpha$.

A.1. A mapping $f$ from E into F is called a multihomogeneous polynomial of multidegree $\alpha$ (with $\alpha \in \mathbf{N}^n$) on E with values in F if there exists an element

such that $f = u \circ p_\alpha$.

A.2. We denote by $P_\alpha(E_1, \ldots, E_n; F)$ the image of $\mathscr{L}_\alpha(E_1, \ldots, E_n; F)$ by the linear mapping $u \mapsto u \circ p_\alpha$ from $\mathrm{Hom}_\alpha(E_1, \ldots, E_n; F)$ into the space of mappings from E into F, endowed with the quotient topology of that of $\mathscr{L}_\alpha(E_1, \ldots, E_n; F)$. An element of $P_\alpha(E_1, \ldots, E_n; F)$ is called a continuous multihomogeneous polynomial of multidegree $\alpha$ on E with values in F. The topology of $P_\alpha(E_1, \ldots, E_n; F)$ is defined by the family of semi-norms:

for $y$ describing the set of continuous semi-norms on F. If F is a normed space, with norm $\gamma$, we write $\|f\|$ instead of $\|f\|_y$. The space $P_\alpha(E_1, \ldots, E_n; F)$ and its topology do not change if one substitutes equivalent norms for the given norms on each $E_i$. They can therefore be defined when the $E_i$ are normable topological vector spaces.

In particular, an element of $P_k(E; F)$ is called a continuous homogeneous polynomial of total degree $k$ on E with values in F. The space $P_k(E; F)$ is the topological direct sum of the spaces $P_\alpha(E_1, \ldots, E_n; F)$ for $|\alpha| = k$. The space $P_0(E; F)$ is the space of constant mappings from E into F and is identified with F.

A.3. We denote by $P(E; F)$ or $P(E_1, \ldots, E_n; F)$ the vector subspace of the vector space of all mappings from E into F generated by the subspaces $P_k(E; F)$.

It is the direct sum of the subspaces $P_\alpha(E_1, \ldots, E_n; F)$ for $\alpha \in \mathbf{N}^n$, and also of the subspaces $P_k(E; F)$ for $k \in \mathbf{N}$. An element of $P(E; F)$ is called a continuous polynomial on $E$ with values in $F$.

A.4. Let $G_j$ be normed spaces (for $1 \leq j \leq m$). Let $f_j \in P(E_1, \ldots, E_n; G_j)$ (for $1 \leq j \leq m$) and let $g \in P(G_1, \ldots, G_m; F)$. The mapping

A.5. We denote by $\hat{P}(E_1, \ldots, E_n; F)$ the product vector space of the $P_\alpha(E_1, \ldots, E_n; F)$ for $\alpha \in \mathbf{N}^n$, endowed with the product topology of the discrete topologies on each of the factors. This topology makes $\hat{P}(E_1, \ldots, E_n; F)$ a separated and complete topological group. The linear mapping from $P(E_1, \ldots, E_n; F)$ into $\hat{P}(E_1, \ldots, E_n; F)$ extending the canonical injections of the $P_\alpha(E_1, \ldots, E_n; F)$ is injective and its image is dense in $\hat{P}(E_1, \ldots, E_n; F)$: one generally identifies a continuous polynomial on $E$ with values in $F$ with its image in $\hat{P}(E_1, \ldots, E_n; F)$.

The identical mapping of $P(E_1, \ldots, E_n; F) = P(E; F)$ extends by continuity to an isomorphism of $\hat{P}(E_1, \ldots, E_n; F)$ onto $\hat{P}(E; F)$.

An element of $\hat{P}(E_1, \ldots, E_n; F)$ is called a formal series with continuous components (or by abuse of language a formal series) on the product of the $E_i$ with values in $F$.

A.6. Let $G_j$ (for $1 \leq j \leq m$) be normed spaces and let $g \in P(G_1, \ldots, G_m; F)$. The mapping $f \mapsto g \circ f$ from $\prod_{1 \leq j \leq m} P(E_1, \ldots, E_n; G_j)$ into $P(E_1, \ldots, E_n; F)$ extends by continuity to a mapping (again denoted $f \mapsto g \circ f$) from $\prod_{1 \leq j \leq m} \hat{P}(E_1, \ldots, E_n; G_j)$ into $\hat{P}(E_1, \ldots, E_n; F)$.

Let $f_j = (f_{j,\alpha})_{\alpha \in \mathbf{N}^n} \in \hat{P}(E_1, \ldots, E_n; G_j)$, with $f_{j,0} = 0$. Put $f = (f_j)$: the mapping $g \mapsto g \circ f$ from $P(G_1, \ldots, G_m; F)$ into $\hat{P}(E_1, \ldots, E_n; F)$ extends by continuity to a mapping (again denoted $g \mapsto g \circ f$) from $\hat{P}(G_1, \ldots, G_m; F)$ into $\hat{P}(E_1, \ldots, E_n; F)$.