- Thread starter
- #1

I need help in order to fully understand Tu's section on the multilinear functions and k-tensors... ...

Section 3.3 reads as follows:

At the beginning of Section 3.3: Multilinear Functions, Tu writes the following:

" ... ... Denote by \(\displaystyle V^k = V \times \ ... \ \times V\) the Cartesian product of \(\displaystyle k\) copies of a real vector space \(\displaystyle V\). A function \(\displaystyle f \ : \ V^k \to \mathbb{R}\) is \(\displaystyle k\)-linear ... ... "

... ... then near the end of Section 3.3 we read ... :

" ... ... We are especially interested in the space \(\displaystyle A_k(V)\) of all alternating \(\displaystyle k\)-linear functions on a vector space \(\displaystyle V\) for \(\displaystyle k \gt 0\). ... ... "

My question is the following: Is the vector space \(\displaystyle V\) in \(\displaystyle A_k(V)\) over the reals ... and further are all the functions involved real-valued ... and is this true in Tu's ongoing treatment multilinear functions and tensors ... ... ?

Hope that someone can help ... ...

Peter