Tensor product of tensors

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Definition

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

• $(f\otimes g)(u_1,\ldots,u_k,v_1,\ldots,v_\ell):\eq f(u_1,\ldots,u_k)\cdot g(v_1,\ldots,v_\ell)$

To call $f\otimes g$ a tensor we require:

• $f\otimes g$ to be multilinear map

This is easily seen

Properties

Let $f$, $g$ and $h$ be "compatible" tensors. Then:

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

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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 $\otimes$ is a nice way to write a product, a nice way to write $\phi_I$.

Notes

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

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

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