Lemma 3

Let $(a_n)_{n\geqslant 0}$ be a sequence of positive real numbers such that the power series

has a finite radius of convergence $r >0$. Let $D\subset \mathbf{C}$ be the open disc with center 0 and radius $r$. There does not exist a holomorphic function $\widetilde{f}$ defined on an open neighbourhood U of $r$ in $\mathbf{C}$ which coincides with $f$ on $U\cap D$.

Suppose there exists such a holomorphic function $\widetilde{f}$ defined on an open neighbourhood U of $r$. There exist real numbers $s < r$ and $\delta >0$ such that $s+\delta > r$ and such that the open disk $D'$ with center $s$ and radius $\delta$ is contained in U. The power series expansion of $\widetilde{f}$ at the point $s$ (VAR, R1, p. 26, 3.2.1) then converges for every $z$ in the disk with center 0 and radius $\delta$ (VAR, R1, p. 29, 3.3.4). This series is

$+\infty (n)+\infty (n)$

$\widetilde{f}_s(z) =\sum\widetilde{f}n$!$(s)z^n=\sum fn$!$(s)z^n$

$n=0n=0$ (VAR, R1, p. 27, 3.2.4). Since $s\in D$, we have

(VAR, R1, p. 28, 3.2.11). Let us take $z$ such that $0< z < \delta$. Since $a_k\geqslant 0$, we have

$\widetilde{f}_s(z) =\sum^{+\infty}(\sum^{+\infty}k(k-1)\cdots n($!$k-n+ 1)a_ks^{k-n})z^n$

$n=0k=n$ $+\infty k+\infty$

$=\sum(a_k\sum n$!($kk-$! $n$)! $s^{k-n}z^n)=\sum a_k(s+z)^k$

$k=0n=0k=0$

(TG, III, p. 40). When $s+z > r$, the convergence of this expression contradicts the assumption that the radius of convergence of the power series $f$ is equal to $r$.

Let E be a real Banach space. Let C be a pointed convex cone in E. The vector space generated by C is equal to $C-C$ (EVT, II, p. 12, cor. 1). In particular, the cone C is total in E if and only if $C-C$ is dense in E. Recall that C is said to be salient if $C\cap (-C)$ is reduced to 0 (EVT, II, p. 11).

The polar $C^{\circ}$ of C is the set of continuous linear forms $\ell \in E'$ such that $\ell (x)\geqslant 0$ for all $x\in C$ (EVT, II, p. 47, prop. 4). By the bipolar theorem (EVT, II, p. 48, th. 1), if C is a closed pointed convex cone, one has $C^{\circ \circ}= C$.