criteria for existence of antidervatives

Let $X$ be a normed space, $Y$ a Banach space, $U\\subset X$ a connected open set, $f\\colon U\\to L(X;Y)$ a continuous function, where $L(X;Y)$ is the space of continuous linear operators. In this article a path is a curve that has bounded variation. The following theorems give necessary and sufficient conditions for $f$ to have an antiderivatives.

Theorem 1.

The following conditions are equivalent:

1.
$f$ has an antiderivative on $U$ ,
2.
for any $\\gamma$ closed path in $U$ $\\int_{\\gamma}f=0$ ,
3.
for any $\\gamma$ , $\\delta$ paths in $U$ that have the same starting and endpoints $\\int_{\\gamma}f=\\int_{\\delta}f$ .

The next theorem states criteria for the existence of local antiderivatives.

Theorem 2.

The following conditions are equivalent:

1.
$f$ has an antiderivative locally,
2.
for $\\gamma$ , $\\delta$ homotopic closed paths in $U$ $\\int_{\\gamma}f=\\int_{\\delta}f$ ,
3.
if $\\gamma$ is a triangular path such that its convex hull is in $U$ , then $\\int_{\\gamma}f=0$ .

With the stronger assumption that $f$ is differenciable we can obtain a more easily applicable condition. We introduce the canonical isometric isomorphism

\\pi_{1,1}\\colon L(X;L(X;Y))\\to L_{2}(X;Y),\\quad u\\mapsto((x_{1},x_{2})\\mapsto u%(x_{1})(x_{2}))

where $L_{2}(X;Y)$ is the space of bilinear operators from $X$ to $Y$ . If $F$ is an antiderivative of $f$ , then $\\pi_{1,1}(Df(x))=D^{2}F(x)$ and by Clairaut’s theorem the second derivative is symmetric. The following theorems assert that the reverse is also true.

Theorem 3.

If $f$ is differentiable, then it has an antiderivative locally if and only if $\\pi_{1,1}(Df(x))$ is symmetric for all $x\\in U$ .

Combining these three theorems immediately gives the following.

Corollary 1.

If $U$ is simply connected and $f$ is differentiable, then it has an antiderivative on $U$ if and only if $\\pi_{1,1}(Df(x))$ is symmetric for all $x\\in U$ .