Show that a (1,2)-tensor is a linear function

[KungFu]

Homework Statement

problem:
1a) show that a (1,2)-tensor T is a linear function T: $E^* x E x E$
(A (1,2)-tensor is a tensor that takes one vector from E and two covectors form the dual space #E^*#

Relevant Equations

I need some help on where to start.

I know that a tensor can be seen as a linear function.
I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.

 

[fresh_42]

Your explanation of $(1,2)$ is ambiguous. Is it an element of $E\otimes E^*\otimes E^*$ or an element of $E^*\otimes E\otimes E$? Let us assume we have a tensor $T\in E\otimes E^*\otimes E^*$. This means we have
$$
T=\sum_{\rho=1}^r U_\rho \otimes v_\rho \otimes w_\rho\; , \; (X,Y) \longmapsto \sum_{\rho=1}^r U_\rho \cdot v_\rho(X) \cdot w_\rho(Y) \in E
$$
What kind of function is $T$ then? The situation for $T'\in E^*\otimes E \otimes E$ is accordingly.

 

[KungFu]

Your explanation of $(1,2)$ is ambiguous. Is it an element of $E\otimes E^*\otimes E^*$ or an element of $E^*\otimes E\otimes E$? Let us assume we have a tensor $T\in E\otimes E^*\otimes E^*$. This means we have
$$
T=\sum_{\rho=1}^r U_\rho \otimes v_\rho \otimes w_\rho\; , \; (X,Y) \longmapsto \sum_{\rho=1}^r U_\rho \otimes v_\rho(X) \otimes w_\rho(Y) \in E
$$
What kind of function is $T$ then? The situation for $T'\in E^*\otimes E \otimes E$ is accordingly.

yes, sorry, I had a typo in my description. it is an element of $E\otimes E^*\otimes E^*$

Can you please explain to me what the capital U and the w and v are in your expression? and also the X and the Y

 

[fresh_42]

The capital letters $U_\rho,X,Y \in E$ are vectors, and the small $v_\rho,w_\rho \in E^*$ are covectors, i.e. linear forms.

Here is a general description:
https://www.physicsforums.com/insights/what-is-a-tensor/

 

[fresh_42]

yes, sorry, I had a typo in my description. it is an element of $E\otimes E^*\otimes E^*$

Can you please explain to me what the capital U and the w and v are in your expression? and also the X and the Y

The tensor isn't a linear function in that case. What is it?

 

[KungFu]

The tensor isn't a linear function in that case. What is it?

The tensor is a mulitlinear function, but why?, I don't see the arguments. I have been told that it is linear in each variable, due to the distributivity of the field R (real numbers), but can I show it, or is it just by definition ?

 

[fresh_42]

You can show it with the definition, see my corrected post #2. I made a copy and paste error and forgot to change $\otimes$ into $\cdot$ which I now did.