I reading john Lee's book entitled "Introduction to Smooth Manifolds" and on page 311
$$T^{k}\!\!\left(V^{*}\right)=V^{*}\!\!\times\!\!V^{*}...V^{*}\;\;k\text{-times}$$
is defined as the vector space of all covariant $k$-tensors on $V$. The stared version of $V$ seems to appear out of nowhere, so my questions are:
What does $V^{*}$ represent?
What is the "*" in $V^{*}$ supose to represent?
Is it true that the vector space $V$ must first be specified before $V^{*}$ can be constructed?
What does an element of $V^{*}$ look like and how does it compare to and element of $V$?
The ${}^*$ in $V^*$ means "dual". The space $V^*$ is the dual space of $V$, which consists of all linear functionals on $V$. Linear functionals on $V$ are linear maps $F : V\to \Bbb R$.