About two definitions of tensor product. (in James R. Munkres's book and Ichiro Satake's book) I am reading "Analysis on Manifolds" by James R. Munkres.  

Definition:
  Let $f$ be a $k$-tensor on $V$ and let $g$ be an $l$-tensor on $V$. We define a $k+l$ tensor $f \otimes g$ on $V$ by the equation $$(f \otimes g)(v_1, \cdots, v_{k+l}) = f((v_1, \cdots, v_{k}) \cdot g(v_{k+1}, \cdots, v_{k+l}).$$ 
It is easy to check that the function $f \otimes g$ is multilinear; it is called the tensor product of $f$ and $g$.  

In the above definition, the operands for $\otimes$ are tensors on $V$.  
When I checked "Linear Algebra" by Ichiro Satake, I found the operands for $\otimes$ are vector spaces.
What is the relation between these two $\otimes$s?
 A: If $V$ and $W$ are vector spaces, their tensor product is a vector space $V \otimes W$ with the following (informally stated) property: 

$(\star)$ Linear maps $V \otimes W \to U$ are in bijective correspondence with bilinear functions $V \times W \to U$. 

The motivation for this is to encode multilinear maps as linear maps, which we understand thoroughly.
Moreover, we have
$$
(V \otimes W)^\ast \simeq \hom(V \otimes W, \Bbbk) \simeq \operatorname{Bil}(V \times W,\Bbbk).
$$
If $f_1, \dots, f_n$ and $g_1, \dots, g_m$ are dual basis of basis $\{v_i\}$ and $\{w_i\}$ in $V$ and $W$, in particular they define linear functions $V \times W \to \Bbbk$ which I will denote with the same letters. For example $f_1$ induces the map $(v,w) \mapsto f_1(v)$.
Given $f_i$ and $g_j$, we can also define the bilinear function
$$
f_i \otimes g_j(v,w) = f_i(v)g_j(w).
$$
We will call this the tensor product of $f_i$ and $g_j$ because of their relation with respect to the roles of $V$ and $W$ in their tensor product,

Proposition: the set $\{f_i \otimes g_j\}_{i,j}$ is a basis for $(V \otimes W)^\ast$. 

Proof. Linear independence comes from the fact that
$$
f_i \otimes g_j(v_k,w_l) = \delta_{ik}\delta_{jl}.
$$
If $d : V \times W \to \Bbbk$ is bilinear and $x = \sum a_iv_i, y = \sum b_iw_i$, then 
$$
d(x,y) = \sum_{i,j}a_ib_jd(v_i,w_j)
$$
and thus $d$ only depends on its values in each pair $(v_i,w_j)$. Noting $d_{ij} = d(v_i,w_j)$ and writing $\varphi = \sum_{i,j}d_{ij}f_i \otimes g_j$, we thus obtain that $d = \varphi$ since $d(v_i,w_j) = \varphi(v_i,w_j)$ for each $i,j \ \square$. 
In the same fashion, fixing a vector space $V$ we can consider the $n$ (tensor) powers of $V$,
$$
V^{\otimes n} := V \otimes \cdots \otimes V,
$$
whose dual space turns out to be the space of  $n$-multilinear maps $V^n \to \Bbbk$. Now, in general, if $v \in V^{\otimes k \ast}$ is a $k$-multiniear function and $w \in V^{\otimes l \ast}$ is an $l$-multilinear one, we can form their tensor product 
$$
v \otimes w(v_1, \dots, v_k,{v'}_1,\dots,{v'}_l) = v(v_1,\dots,v_k)w({v'}_1,\dots,{v'}_l)$ 
$$
which is an element of ${V^{\otimes (k+l)}}^\ast$.
