Tensor product of real spaces I have just begun learning about tensor products. Given are the following relations of the tensor product between vector spaces $A$ and $B$:


*

*$\lambda(a\otimes b)=(\lambda a)\otimes b=a\otimes(\lambda b) $ for $\lambda \in \mathbb{R}$.

*$a\otimes b_1 + a\otimes b_2 = a\otimes(b_1+b_2)$.

*$a_1\otimes b + a_2\otimes b=(a_1+a_2)\otimes b$.


For $a\in A$ and $b\in B$.
Now I want to show that $\mathbb{R}\otimes\mathbb{R}\cong \mathbb{R}$, but I am not sure where to start. Is it necessary to explicitly construct an isomorphism between the two spaces? Is there something else I should know about tensor products to get this started?
 A: Just consider the map$$\begin{array}{ccc}\mathbb{R}\bigotimes\mathbb{R}&\longrightarrow&\mathbb R\\\displaystyle\sum_{k=1}^na_k\otimes b_k&\mapsto&\displaystyle\sum_{k=1}^na_kb_k.\end{array}$$It is an isomorphism.
A: $\dim A\otimes B=\dim A\dim B$ so when $\dim A=1$ and $\dim B$ is finite then $A\otimes B\cong B$, for there are no two distinct linear spaces of finite dimension over the same field, up to isomorphisms.
If you do not want to use dimensions, the next is another proof, equally obvious.
If $A$ is a field and $B$ a linear space of finite dimension over $A$ the multiplication of a vector by a scalar defined in $B$ satisfies the axioms $1$-$3$. That means by definition of tensor product that there exists and is unique a homomorphism of linear spaces $\eta:A\otimes B\rightarrow B$ such that $\eta(a\otimes b)=ab.$ You now see that in this case it is also an isomorphism. Indeed it is surjective because $\forall x\in B,~x=1x=\eta(1\otimes x)$, and it is injective because if $\sum_i a_i\otimes b_i\ne \sum_i c_i\otimes d_i$, that means that there are no relations among axioms $1$-$3$ through which equality can be proved, then, being the axioms of the multiplication of a vector by a scalar formally the same as axioms $1$-$3$, there are no relations that can prove the equality of $ab$ and $cd$, that means that $\eta(\sum_i a_i\otimes b_i)\ne \eta(\sum_i c_i\otimes d_i)$.  
