Tensor product of tensors

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Definition

Let [ilmath]f[/ilmath] and [ilmath]g[/ilmath] be "tensors of rank [ilmath]k[/ilmath] and [ilmath]\ell[/ilmath] respectively", then we can define the tensor product, a rank [ilmath]k+\ell[/ilmath] tensor, as follows:

• [ilmath](f\otimes g)(u_1,\ldots,u_k,v_1,\ldots,v_\ell):\eq f(u_1,\ldots,u_k)\cdot g(v_1,\ldots,v_\ell)[/ilmath]

To call [ilmath]f\otimes g[/ilmath] a tensor we require:

• [ilmath]f\otimes g[/ilmath] to be multilinear map

This is easily seen

Properties

Let [ilmath]f[/ilmath], [ilmath]g[/ilmath] and [ilmath]h[/ilmath] be "compatible" tensors. Then:

1. Associativity: [ilmath]f\otimes(g\otimes h)\eq(f\otimes g)\otimes h[/ilmath]
2. Homogeneity: [ilmath](cf)\otimes g\eq c(f\otimes g)\eq f\otimes(cg)[/ilmath] - for scalar [ilmath]c\in\mathbb{F} [/ilmath]
3. Distributivity: (suppose that [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are of the same rank/order (among other things?)) then:
   1. [ilmath](f+g)\otimes h\eq f\otimes h+ g\otimes h[/ilmath]
   2. [ilmath]h\otimes(f+g)\eq h\otimes f+h\otimes g[/ilmath]
4. Suppose that we are dealing with [ilmath]\mathcal{L}^k(V)[/ilmath] in Munkres' notation, or [ilmath]L(\underbrace{V,\ldots,V}_{k\ \text{lots of }V};\mathbb{F})[/ilmath] in mine for vec space [ilmath](V,\mathbb{F})[/ilmath] of dimension [ilmath]n[/ilmath], then:
   • If [ilmath]e_1,\ldots,e_n[/ilmath] is a basis of [ilmath]V[/ilmath] we have:
     • The corresponding "elementary tensors", [ilmath]\phi_I[/ilmath] satisfy the equation:
       • [math]\phi_I\eq\phi_{i_1}\otimes\phi_{i_2}\otimes\cdots\otimes\phi_{i_{k-1} }\otimes\phi_{i_k} [/math] where [ilmath]I\eq(i_1,\ldots,i_k)[/ilmath] and each [ilmath]i_j\in\{1,\ldots,n\} [/ilmath]

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Easy to do, I've done most of this on paper already.

• Property 4 is easier and more significant than I first thought. It just shows [ilmath]\otimes[/ilmath] is a nice way to write a product, a nice way to write [ilmath]\phi_I[/ilmath].

Notes

• [ilmath]\phi_i(e_j):\eq\left\{\begin{array}{lr}1 & \text{if }i\eq j\\ 0 & \text{otherwise} \end{array}\right.[/ilmath] - these align with dual basis vectors
• [ilmath]\phi_I(e_{j_1},\ldots,e_{j_k}):\eq\left\{\begin{array}{lr}1 & \text{if }J\eq I\\ 0 & \text{otherwise}\end{array}\right.[/ilmath]
  • [ilmath]\phi_I(e_{j_1},\ldots,e_{j_k})\eq\phi_{i_1}(e_{j_1})\cdot\ \ldots\ \cdot\phi_{i_k}(e_{j_k})[/ilmath] follows easily.
    • where [ilmath]I[/ilmath] and [ilmath]J[/ilmath] are both [ilmath]k[/ilmath]-tuples of elements of [ilmath]\{1,\ldots,n\} [/ilmath]

These are the elementary tensors. Tensors of the form [ilmath]\phi_I[/ilmath] form a basis of [ilmath]L(\underbrace{V,\ldots,V}_{k\text{ lots of }V};\mathbb{F})[/ilmath]

This is easy to see, see page 222 of Munkres on manifolds for details.