Proposition 5

The presheaf $\mathscr{D}'(U)$ is a sheaf.

Let us recall (TA, I, p. 43, def. 2) that this means that for every open subset V of U and every open covering $(V_i)_{i\in I}$ of V, the following conditions are satisfied:

(i) The mapping $(r_{V_iV})_{i\in I}:\mathscr{D}'(V)\rightarrow \prod_{i\in I}\mathscr{D}'(V_i)$ is injective;

(ii) For every family $(f_i)\in \prod_{i\in I}\mathscr{D}'(V_i)$ such that

for every pair $(i, i')\in I\times I$, there exists a distribution $f\in \mathscr{D}'(V)$ such that for every $i\in I$, one has $r_{V_iV}(f) =f_i$.

Let V be an open set of U and $\mathscr{V}= (V_i)_{i\in I}$ an open covering of V. Let $(W_j)_{j\in J}$ be a locally finite open covering of V, finer than $\mathscr{V}$ and formed of relatively compact subsets (lemma 2 of IV, p. 201). Let us fix a partition of unity $(\varphi_j)_{j\in J}$ subordinate to the covering $(W_j)_{j\in J}$ such that the support of $\varphi_j$ is contained in $W_j$ for every $j\in J$ (lemma 1 of IV, p. 196).

Let $\varphi \in \mathscr{D}(V)$. Since the set of $j\in J$ such that $W_j$ meets the support of $\varphi$ is finite (TG, I, §9, n$^o1$, p. 59), one has

where the sum consists of only a finite number of non-zero terms.

Let us prove (i). Suppose that $f\in \mathscr{D}'(V)$ satisfies $r_{V_iV}(f) = 0$ for every $i\in I$. This means that $\langle f, \varphi \rangle = 0$ for every test function $\varphi$ whose support is contained in one of the open sets $V_i$. This is fortiori true if the support of $\varphi$ is contained in one of the open sets $W_j$. But then, for every $\varphi \in \mathscr{D}$(V), we have

hence $f= 0$.

Let us prove (ii). Let $(f_i)_{i\in I}$ be a family such that $f_i\in \mathscr{D}'(V_i)$ for every $i\in I$ and $r_{V_i\cap V_{i'},V_i}(f_i) =r_{V_i\cap V_{i'},V_{i'}}(f_{i'})$ for all $i$ and $i'$ in I. Let $\iota : J\rightarrow I$ be a mapping such that $W_j\subset V_{\iota(j)}$ for every $j\in J$. For every $j\in J$, put $\widetilde{f}_j=r_{W_j,V_{\iota(j)}}(f_{\iota(j)})$. We then have

Exchanging the role of $j$ and $j'$ and noting that

by assumption, we deduce that

for all $(j, j')\in J^2$.

For $\varphi \in \mathscr{D}$(V), put

where the sum is finite since only a finite number of terms can be non-zero.

The mapping $\lambda$ is a linear form on $\mathscr{D}(V)$. Let us prove that it is a distribution. Let B be a bounded subset of $\mathscr{D}(V)$ and let K be a compact subset of V such that $B\subset \mathscr{C}_K^{\infty}(V)$. By TG, I, §9, n$^o1$, p. 59, there exists a finite subset $J'$ of J such that

for every $\varphi \in B$. Since $\widetilde{f}_j$ is a distribution for every $j\in J'$ and $\mathscr{D}(V)$ is a topological algebra, we deduce that $\lambda$ is bounded on B, whence the result (lemma 4).

Let $j\in J$ and let $\varphi \in \mathscr{D}(V)$ whose support is contained in $W_j$. We then have

by (1), since $\varphi \varphi_{j'}$ has support contained in $W_j\cap W_{j'}$. Consequently

whence $r_{W_jV}(\lambda ) =\widetilde{f}_j$ for every $j\in J$.

Let $i\in I$. Let us show finally that the restriction of $\lambda$ to $V_i$ coincides with $f_i$. By condition (i), applied to the covering of $V_i$ by the open sets $W_j$, it is enough to verify that for every $j\in J$, the restriction of $\lambda$ to $V_i\cap W_j$ coincides with that of $f_i$. By what precedes, it is a question of verifying that the restriction of $f_i$ to $V_i\cap W_j$ is that of $\widetilde{f}_j$. Now one has

where one has used the assumption concerning the family $(f_i)$. But then

which permits one to conclude.