1
$\begingroup$

Let V be a finite dimensional real vector spaces of dimension n with basis $e_i$, $1 \leq i \leq n$. We denote by $x_i$ the coordinate function, namely the function that associate to $v = v_1e_1+...+v_ne_n \in V $ the element $v_i$.
Let P be another finite dimensional real vector space. We set $W= V \oplus P $ and denote by $C^{-\infty}(W)$ the set of generalized function over W.

I find the following statement in the proof of lemma 21 in the article by Kumar and Vergne on equivariant cohomology with generalized coefficients :

If $x_if(x+y)=0$, for all i. Then f is the product of the $ \delta$-function on V with a generalized function on P .

Why this is true?

Any help would be greatly appreciated.

$\endgroup$
2
  • 2
    $\begingroup$ First get rid of the vector space language and symbols like $\oplus$ etc. This is just multivariate calculus, but with a distributional twist. You have a distribution $f(x_1,\ldots,x_n,y_1,\ldots,y_p)$ in $n+p$ variables which satisfies $x_i f(x_1,\ldots,x_n,y_1,\ldots,y_p)=0$ for for all $i$, $1\le i\le n$. First look at $n=1$ and $p=0$. This is done in math.stackexchange.com/questions/3172353/… then add more variables and see how to adapt that method. $\endgroup$ Commented Mar 1, 2021 at 23:12
  • $\begingroup$ Hi! I'm interested in this question. Can anyone please one explain to me what does it mean that $f$ is the product of the $\delta-$function on $V$ with a generalized function on $P$, how is this product defined? $\endgroup$ Commented Sep 3, 2021 at 23:30

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.