# Multilinear Functions and Alternating k-tensors ... ... Tu, Section 3.3

#### Peter

##### Well-known member
MHB Site Helper
I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ...

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

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

#### GJA

##### Well-known member
MHB Math Scholar
Hi Peter ,

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 ... ... ?
Yes, $V$ is a vector space over the reals and $f$ is real-valued. Tu uses the word function to mean real-valued; he refers to functions between manifolds as mappings. Though it has been a few years since I read his text, I recall that these conventions are strictly adhered to throughout the book.

#### Peter

##### Well-known member
MHB Site Helper
Hi Peter ,

Yes, $V$ is a vector space over the reals and $f$ is real-valued. Tu uses the word function to mean real-valued; he refers to functions between manifolds as mappings. Though it has been a few years since I read his text, I recall that these conventions are strictly adhered to throughout the book.

Thanks GJA ... appreciate your help ...

Peter