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. 1.

$f$ has an antiderivative on $U$,

2. 2.

for any $\gamma$ closed path in $U$ $\int_{\gamma}f=0$,

3. 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. 1.

$f$ has an antiderivative locally,

2. 2.

for $\gamma$, $\delta$ homotopic closed paths in $U$ $\int_{\gamma}f=\int_{\delta}f$,

3. 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 $x\in U$.

Title criteria for existence of antidervatives CriteriaForExistenceOfAntidervatives 2013-03-22 19:14:00 2013-03-22 19:14:00 scineram (4030) scineram (4030) 9 scineram (4030) Theorem msc 46G05