Suppose $F$ is an equivalence of the categories $Vect_{k}$ - the category of all $k$ vector spaces where $k$ is a field - and its opposite category: $Vect_{k}^{op}$. I have shown that the (unique up to isomorphism) 1-dimensional vector space $k$ gets sent to itself under $F$. I need to show this gives a contradiction. As a hint I am given that the dual of an infinite dimensional vector space has a higher dimension (in the sense of cardinal arithmetic) than the vector space itself.
My best guess on how to proceed is using that any (infinite-dimensional) vector space is the direct sum (coproduct) of copies of $k$ and an equivalence (in particular a left adjoint) will preserve coproducts and thus send the vector space to itself. I know that in the case of the category of finite-dimensional $k$-vector spaces an equivalence is given by sending a vector space to its dual and that all equivalences are unique up to natural isomorphism so perhaps if such an equivalence $F$ were to exist, it must send vector spaces to their dual and thus we get the required contradiction.
EDIT: A coproduct of copies of $k$ in $Vect_{k}$ gets sent to a coproduct of copies of $k$ in the opposite category, but as an abstract vector space this will be the product of copies of $k$.