We define a Projective space of a vector space as follow : http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces
Given a vector space $V$ over a field $\mathbb{K}$, its associated projective space $\mathbb{P}(V)$ is by definition :
$\mathbb{P}(V) = V - \{0\}/\sim$ where $\sim$ is the equivalence relation
$u \sim v$ iff $u = \lambda v$ for $u, v\in V - \{0\}$ and $\lambda \in \mathbb{K}$.
In particular,
The real projective space $\mathbb{R} P^n$ is the quotient space of $\mathbb{R}^{n+1}-\{0\}$ by the equivalence relation defined on $\mathbb{R}^{n+1}-\{0\}$ by
$$x\sim y \iff y=tx$$
for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$.
Geometrically, two nonzero points in $\mathbb{R}^{n+1}$ are equivalent if and only if they lie on the same line through the origin, so $\mathbb{R} P^n$ can be interpreted as the set of all lines through the origin in $\mathbb{R}^{n+1}$.
In other hand, For two real vector spaces $V$ and $W$, the Tensor-Product $V \otimes W$ is the Quotient space $K^{V\times W} / R(V,W)$, where $R(V,W)$ is generated by the vectors \begin{align*} (v,\lambda w) - \lambda (v,w) & \\ (v,w+w')-(v,w)-(v,w') & \\ (\lambda v,w) - \lambda(v,w) & \\ (v+v',w)-(v,w)-(v',w) & \\ \text{and} \quad K^{V\times W} := \{f: V\times W \rightarrow\ K\}. & \end{align*}
for any basis $(v_i)_{i\in I}$ of V and $(w_j)_{j\in J}$ of W, $(v_i\otimes w_j)_{i\in I,j\in J}$ is a basis of $V\otimes W$, so any $a\in V\otimes W$ is a linear combination of some vectors $v_i\otimes w_j$.
And a tensor of type $(n, m)$ is an element of $V^{\ast \otimes m} \otimes V^{\otimes n}$.
For example,
- A tensor of type $(0, 0)$ is a scalar.
- A tensor of type $(1, 0)$ is a vector.
How can Intemperate geometrically the tensor product of two projectifs space ? What is interpretation of the tensor product of two Grassmannians ?
Thanks in advance.
