Tensor product of vector spaces
Finite dimensional case
Let $V,U$ be finite dimensonal vector spaces over $K$.
Let $Free(V,U)$ be the set of finite linear combinations $$\sum_{i=1}^na_i(x_i,y_i)$$ where $n\in\mathbb{Z}+$, $x_i\in V$, $y_i\in U$, and $a_i\in K$.
Definition. The tensor product of $V$ and $U$, denoted $V\otimes U$, is the quotient $Free(V,U)/\sim$, where $\sim$ is the bilinear equivalence relation: $$ (ax+bz,y)=a(x,y)+b(z,y),\quad (x,ay+bt)=a(x,y)+b(x,t). $$
Basis
Proposition. If ${v_1,\dots,v_n}$ is a basis of $V$ and ${u_1,\dots,u_m}$ is a basis of $U$, then $$ {v_i\otimes u_j\mid 1\leq i\leq n, \ 1\leq j\leq m} $$ forms a basis of $V\otimes U$.
Symmetric and antisymmetric tensors
Definition. The subspace of symmetric tensors $S^2(V)\subset V\otimes V$ is defined as $$ \text{span}{x\otimes y + y\otimes x\mid x,y\in V}. $$
Definition. The subspace of antisymmetric tensors $\Lambda^2(V)\subset V\otimes V$ is defined as $$ \text{span}{x\otimes y-y\otimes x\mid x,y\in V}. $$
On elementary tensors, it is obvious we these are named the way they are.
Properties
- If $V$ is finite-dimensional, then $V\otimes V^\ast\cong\text{End}(V)$.
- Given $A\in\text{End}(V)$ and $B\in\text{End}(U)$, we can construct a well defined element $A\otimes B\in\text{End}(V\otimes U)$ by the formula $$ A\otimes B(v\otimes u)=(Av)\otimes (Bu) $$.
- If $V$ is finite dimensional, then $S^2(V)\oplus\Lambda^2(V)=V\otimes V$.