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In this paper properties of field functionals and characterization of local functionals they have at page 5

Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$ $$ D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} . $$

Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.

at page 24 they have

Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that $$ D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x $$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$ is the dual of the space of section $\Gamma (E)$

Since $h(x)$ is a section in Lemma VI.2, how can we produce a number by multiplying it by a function and then intergrate it ?

Am I missing something?

In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions

Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$ $$ D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} . $$
Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.

At page 24 they state the following

Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that $$ D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x $$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$ is the dual of the space of section $\Gamma (E)$

Since $h(x)$ is a section in Lemma VI.2, how can we produce a number by multiplying it by a function and then intergrate it ?

Am I missing something?

In this paper properties of field functionals and characterization of local functionals they have at page 5

Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$ $$ D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} . $$

Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.

at page 24 they have

Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that $$ D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x $$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$ is the dual of the space of section $\Gamma (E)$

Since $h(x) is a section in Lemma VI.2, how can we produce a number by multiplying it by a function and then intergrate it ?

Am I missing something?

In this paper properties of field functionals and characterization of local functionals they have at page 5

Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$ $$ D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} . $$

Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.

at page 24 they have

Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that $$ D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x $$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$ is the dual of the space of section $\Gamma (E)$

Since $h(x)$ is a section in Lemma VI.2, how can we produce a number by multiplying it by a function and then intergrate it ?

Am I missing something?

In this paper properties of field functionals and characterization of local functionals they have at page 5

Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$ $$ D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} . $$

Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.

at page 24 they have

Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that $$ D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x $$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$ is the dual of the space of section $\Gamma (E)$

Now in the integral in Lemma VI.2. $h(x)$ is a section so how can we produce a number by multiply it by a function and then intergrate . Am I missing something?

In this paper properties of field functionals and characterization of local functionals they have at page 5

Definition II.2. Let $U$ be an open subset of a Hausdorff locally convex space $E$ and let $f$ be a nap from $U$ to a Hausdorff locally convex space $F$. Then $f$ is said to have a derivative at $x \in U$ in the lirection of $v \in E$ if the following limit exists: ${ }^{.3}$ $$ D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} . $$

Definition II.3. Let $U$ be an open subset of a Hausdorff locally convex space E and let fe a map from $U$ to a Hausdorff locally convex space $F$. Then $f$ is Bastiani differentiable on $U$ [denoted by $\left.f \in C^1(U)\right]$ iff has a Gâteaux differential at every $x \in U$ and the map $D f: U \times E \rightarrow F$ defined by $D f(x, v)=D f_x(v)$ is continuous on $U \times E$.

at page 24 they have

Lemma VI.2. Let $U$ be an open subset of $C^{\infty}(M)$ and $F: U \rightarrow \mathbb{K}$ be Bastiani smooth. For every $\varphi$ such that the distribution $D F_{\varphi} \in \mathcal{E}^{\prime}(M)$ has an empty wave front set, there exists a unique function $\nabla F_{\varphi} \in \mathcal{D}(M)$ such that $$ D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x $$

Here $\mathcal{D}(M)$ is the space of test function on $M$ and $\mathcal{E}^{\prime}(M)$ is the dual of the space of section $\Gamma (E)$

Since $h(x) is a section in Lemma VI.2, how can we produce a number by multiplying it by a function and then intergrate it ?

Am I missing something?