Welcome to our community

Be a part of something great, join today!

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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,881
Hobart, Tasmania
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... ...

Section 3.3 reads as follows:


Tu - 1 - Start of Section 3.3 ... PART 1 ... .png
Tu - 2 - Start of Section 3.3 ... PART 2 ... .png





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
Jan 16, 2013
249
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
Jun 22, 2012
2,881
Hobart, Tasmania
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