criteria for existence of antidervatives
Let $X$ be a normed space^{}, $Y$ a Banach space^{}, $U\subset X$ a connected open set, $f:U\to L(X;Y)$ 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
$${\pi}_{1,1}:L(X;L(X;Y))\to {L}_{2}(X;Y),u\mapsto (({x}_{1},{x}_{2})\mapsto u({x}_{1})({x}_{2}))$$ 
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 ${\pi}_{\mathrm{1}\mathrm{,}\mathrm{1}}\mathit{}\mathrm{(}D\mathit{}f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$ is symmetric for all $x\mathrm{\in}U$.
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 ${\pi}_{\mathrm{1}\mathrm{,}\mathrm{1}}\mathit{}\mathrm{(}D\mathit{}f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$ is symmetric for all $x\mathrm{\in}U$.
Title  criteria for existence of antidervatives 

Canonical name  CriteriaForExistenceOfAntidervatives 
Date of creation  20130322 19:14:00 
Last modified on  20130322 19:14:00 
Owner  scineram (4030) 
Last modified by  scineram (4030) 
Numerical id  9 
Author  scineram (4030) 
Entry type  Theorem 
Classification  msc 46G05 