Let $V$ be a $n$-dimensional vector space, say $\mathbb{R}^n$.What is $\wedge^{k} V$?
I just started reading about tensor product and wedge product yesterday, and I am using Analysis on Manifold by Munkres and his notations. Currently my understanding is that:
Given a $k$-tensor $f$ in $\mathcal{L}^k(V)$, and a $l$-tensor $g$ in $\mathcal{L}^l (V)$. We define the tensor product of $f$ and $g$ as a new $(l+m)$-tensor, such that $f\otimes g (v_1,\cdots,v_k,v_{k+1},\cdots,v_{k+l})=f(v_1,\cdots,v_k)g(v_{k+1},\cdots,v_{k+l})$
Similarly, if we only focus on alternating tensors, then we can define the wedge product between two alternating tensors. Let $f\in \mathcal{A}^k (V)$ and $g\in \mathcal{A}^k (V)$, then $f\wedge g$ will be in $\mathcal{A}^{k+l} (V)$ satisfying some properties, such as $f\wedge g = (-1)^{kl} g\wedge f$
But my book doesn't seem to cover wedge product of a vector space. So I read Wiki, and it defines $\bigwedge V$ in terms of tensor algebra of $V$, and found the following:
Let $V$ be a vector space over a field $K$. For any nonnegative integer k, we define the $k$-th tensor power of $V$ to be the tensor product of $V$ with itself k times:
$T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V$
That is, $T^k V$ consists of all tensors on $V$ of order $k$.
Difference in notations confuse me, because how do you "tensor product" two vector space? I assume that $T^k (V) = \mathcal{L}^k (V)$? And if that's the case, I suppose that $\bigwedge^k V$ is just $\mathcal{A}^k(V)$? Namely the elemens in $\bigwedge^k V$ are just $k$-alternating tensors defined on $V$?