Why algebraic dual space of $V$ is denoted by $(V^*)^{ \otimes n}$?

Question

Why algebraic dual space of $V$ is denoted by $(V^*)^{ \otimes n}$, why not simply by $V^*$ ?, $n=\text{dim} \ V$.

What is the need of tensor product here and what does mean it?

Answer 1

You should be careful with content from Wikipedia, some of the articles contain mistakes.

In the present case, $V^\ast$ is the algebraic dual space and $(V^\ast)^{\otimes n}$ denotes the $n$-fold tensor power. What the author of the article means is that $f_1\otimes\cdots\otimes f_n$ is an element of $(V^\ast)^{\otimes n}$.