Consider a finite n-dimensional vector space $V$ over field $\mathbb{k}\in\{\mathbb{R},\mathbb{C}\}$, and let $V^*=\mathcal{L}(V;\mathbb{k})$ be the dual space (linear maps $V\to \mathbb{k}$). I have several questions regarding the definitions and customary notations used for tensors on $V$. (If it matters, I'm primarily interested in the following questions in the context of mechanics/relativity where $V$ is either an n-dimensional euclidean or minkowski space, or the tangent space of a smooth manifold.)
please help answer as many or as few of the following as you like:
Question 1: An (r,s)-tensor is defined as a multilinear map, $T$, that takes in r elements of $V^*$ and s elemts of $V$ to give an element of $\mathbb{k}$. That is, a map $$ T :{V^*}^{\times r}\times V^{\times s}\to \mathbb{k}$$
that is linear in all of it's "slots". If we denote the space of all such tensors as $\mathcal{T}^r_s(V)$, then are the following equalities true?
$$ \mathcal{T}^r_s(V) = V^{\otimes r} \otimes {(V^*)}^{\otimes s} = {\textstyle \bigotimes^r } V {\textstyle \bigotimes^s } V^* $$
are any of the above expressions not equal to the others? I have seen various versions of them appear in different sources and I want to know if they are the same or if any have specific meanings that distinguish them from one another. Additionaly, is it true that $\mathcal{T}^r_s(V) = \mathcal{T}^s_r(V^*)$?
Question 2: If we let $\mathcal{A}^k(V)$ denote the set of all antisymmetric (k,0)-tensors, and we let $\mathcal{A}^k(V^*)$ denote the set of all antisymmetric (0,k)-tensors, then are these simply the same thing as the $k^{th}$ exterior powers of $V$ and $V^*$, repsectivly? That is, are the following relations true:
$$ \textstyle{\bigwedge^k} V = \mathcal{A}^k(V) \;\subset \; \mathcal{T}^k_0(V) \qquad,\qquad \textstyle{\bigwedge^k} V^* = \mathcal{A}^k(V^*) \;\subset \;\mathcal{T}^0_k(V) $$
Specificaly, what is the difference, if any, between $\textstyle{\bigwedge^k} V$ and $\mathcal{A}^k(V)$? Furthermore, is it correct to call an element of $\bigwedge^k V$ a k-vector and an element of $\bigwedge^k V^*$ a k-form? Or do those words mean something else?
Question 3: I wrote $\mathcal{A}^k(V)$ and $\mathcal{A}^k(V^*)$ separatly becuase, for some reeason, I never seem to see mixed-order antisymmetric tensors mentioned. Why not? If you have a non-degenerate metric/inner product on $V$, then couldnt you form an antisymmetric (r,s)-tiensor out of an antisymetric (0,r+s)-tensor? Does it make sense to define some $\mathcal{A}^r_s(V)\subset \mathcal{T}^r_s(V) $ as the set of all antisymmetric (r,s)-tensors? Would it be given by $\mathcal{A}^r_s(V) = \mathcal{A}^r(V) \otimes \mathcal{A}^s(V^*)$? Or would it be $\mathcal{A}^r_s(V) = \mathcal{A}^r(V) \wedge \mathcal{A}^s(V^*)$? Is the exterior product between vector spaces even defined?
Question 4: How does the space of differential k-forms, $\Omega^k(V)$, relate to all of this? Specifically, what is the relation between $\mathcal{A}^k(V^*)$, and $\bigwedge^k V^*$, and $\Omega^k(V)$?
Question 5: Do the answers or definitions of anything above chnge if the vector space $V$ is the tangent space of a smooth manifold?
context: It may be clear from the above that my background is not in math. It is in physics and engineering. I'm currently trying ot teach myself some differential geometry to understand the geometric view of Hamiltonian mechanics. So, in the long run, I want to apply all of the above questions to smooth manifolds where the above $V$ will then be the tangent space of some configuration manifold.
Some Answers:
For the first: There is an isomorphism between the space of multilinear maps and the abstract tensor product (in the case of finite dimensional $V$) $$ \mathcal{T}^r_s(V) \cong V^{\otimes r} \otimes {(V^*)}^{\otimes s} \equiv {\textstyle \bigotimes^r } V {\textstyle \bigotimes^s } V^* $$ The first sign is isomorphism, the second is an equivalence of symbols, i.e. the last two denote the same space. The isomorphism is $$ v_1\otimes .. \otimes v_r \otimes f_1 \otimes .. \otimes f_s \leftrightarrow \left( (h_1,.. ,h_r, u_1, .., u_s) \mapsto h_1(v_1)..h_r(v_r) f(u_1).. f(u_s) \right) $$ extended by linearity. For infinite dimensional the space on the left not isomorphic in this way to that on the right. For the spaces $\mathcal{T}^r_s(V)$ and $\mathcal{T}^s_r(V^*)$, they are also isomorphic, since $V \cong V^{**}$ (in case of finite dimensional $V$) and $X \otimes Y \cong Y \otimes X$.
For the second: the equations should be $\cong$ again. The same difference as above, isomorphism between the absrtact universal exterior algebra and the space of the antisymmetric multilinear maps. Elements of $\Lambda^k V^*$ are called $k$-covectors.
The space $\Omega^k(M)$ is the space of antisymmetric-multilinear-(real-valued-)map-valued fields on the manifold $M$, or simply (differential) $k$-forms. In other words, an element $\omega \in \Omega^k(M)$ is a map from $M$ into the exterior contangent bundle, i.e $\omega: M \ni p \mapsto \omega_p \in \Lambda^k T_p^*M \cong \mathcal{A}^k(T_p^*M) \equiv \mathcal{A}_k(T_pM)$, where $T_pM$ is the tangent space at $p$ and $T_p^*M$ is its dual.
Added:
As the space $\mathcal A^k(V) \subset \mathcal T^k(V)$ the space $\Lambda^k V$ can also be viewed as a subspace of $\bigotimes^k V$, (more precisely embedded in $\bigotimes^k V$). For the subspace interpretation one uses the (linear) antisymmtrisor $$ A: V^{\otimes k} \ni v_1\otimes .. \otimes v_k \mapsto v_1\wedge.. \wedge v_k = \frac{1}{k!}\sum_{\sigma \in S_k} (-1)^\sigma v_{\sigma1}\otimes .. \otimes v_{\sigma k} \in \Lambda^k V $$ to rewrite the elements more compactly. This antisymmetrisor is actually a projector on the subspace $\Lambda^k V$. The correspondence between $\mathcal A^k(V)$ and $\Lambda^k V$ is the restriction of the map between $\mathcal T^k(V)$ and $\bigotimes^k V$ (the first map in my answer).