I will be taking a Differential Geometry class in the Fall, so I decided to get somewhat of a head start by going through Spivak's "Calculus on Manifolds." Before reading, though, I saw the Addenda at the end, which stated that his notation $\Lambda^{k}\left(V\right)$ for the space of alternating $k$-tensors was incorrect, although it is naturally isomorphic to $\Lambda^{k}\left(V^{*}\right)$ for fin. dim. $V$ (and, after a little more digging around on Wikipedia, is naturally isomorphic to $\left(\Lambda^{k}\left(V\right)\right)^{*}$ in general). Is the notation suggested by Spivak, $\Omega^{k}\left(V\right)$, standard or is there some other notation that is typically used?
EDIT: To quote Spivak: "Finally, the notation $\Lambda^{k}\left(V\right)$ appearing in this book is incorrect, since it conflicts with the standard definition of $\Lambda^{k}\left(V\right)$ (as a certain quotient of the tensor algebra of $V$). For the vector space in question (which is naturally isomorphic to $\Lambda^{k}\left(V^{*}\right)$ for finite dimensional vector spaces $V$) the notation $\Omega^{k}\left(V\right)$ is probably on the way to becoming standard."
I don't know if this is still the case, though, or if his use of $\Lambda^{k}\left(V\right)$ became the standard.
Best Answer
In Lee's 'Intro to Smooth Manifolds', $\Lambda^k(V)$ refers to the space of alternating $k$-tensors on a vector space $V$, as you mentioned. However, the space $\Omega^k(M)$ is the space of smooth $k$-forms on a smooth manifold $M$. That is, an element of $\omega \in \Omega^k(M)$ is a smooth map $M \to \Lambda^k(T^* M)$ (called a smooth section of of the bundle $\Lambda^k(T^* M)$), so for each point $x \in M$, we get an alternating $k$-tensor $\omega(x) \in \Lambda^k(T^* M)$. This space is often written as $\Omega^k(M) = \Gamma(\Lambda^k(T^* M))$.
Not entirely sure however what $\Omega^k(V)$ is, when $V$ is just a vector space.