# Partially apply/act on/contract/trace tensor?

Just some definitions to start with: $$\mathcal{Basis}(V_i)=\{e_{(i)1},\dots,e_{(i)n_i}\}\\ \mathcal{Basis}(V^*_i)=\{e^{(i)1},\dots,e^{(i)n_i}\}\\ \mathcal{Basis}(V^*_1\otimes \cdots \otimes V^*_k) = \{e^{(1)b_1}\otimes\cdots\otimes e^{(k)b_k} \equiv \bigotimes_{i=1}^{k}{e^{(i)b_i}}\colon (i \in \{1,\dots k\}) \land (1\leq b_i \leq n_i)\}\\$$ And (not sure which one is "correct"): $$V^*_1\otimes \cdots \otimes V^*_k \ni A = A(e_{(1)b_1},\dots,e_{(k)b_k})\bigotimes_{i=1}^{k}{e^{(i)b_i}}$$ or $$V^*_1\otimes \cdots \otimes V^*_k \ni A = A(e_{(1)b_1}\otimes \dots \otimes e_{(k)b_k})\bigotimes_{i=1}^{k}{e^{(i)b_i}}$$ I wonder if it is possible to "partially apply" arguments to the tensor. I see things like $A(v,-)$ sometimes. Can this be defined in some way to make it easy to "shut up and calculate", using something like this (where $\langle -, -\rangle$ can be chosen to be the natural pairing of the dual space to the space if it so happens that one has chosen $V_{(a)} = V^*$ and $V_{(b)}=V$ below): $$tr_{(a),(b)}(T) = T(e_{(1)b_1}, \dots , e_{(a) b_a} , \dots , e_{(b) b_b} , \dots , e_{(k)b_k})\langle e^{(a)b_a},e^{(b) b_b}\rangle\bigotimes_{i=1\\i\neq a\\i\neq b}^{k}{e^{(i)b_i}}$$ (Note: I'm only kind of sure the above definition of trace is correct) So something like: $$A(v,-) \overset{?}{=} tr_{(1),(3)}(A\otimes v) \overset{?}{=}\\ \overset{?}{=} tr_{(1),(3)}([A(e_{(1)b_1}, e_{(2)b_2})e^{(1)b_1}\otimes e^{(2)b_2}]\otimes [v(e_{(3)b_3})e^{(3)b_3}]) =\langle e^{(1)b_1} , e^{(3)b_3} \rangle \left( [A(e_{(1)b_1}, e_{(2)b_2})e^{(2)b_2}][v(e_{(3)b_3})]\right )\\ =\langle e^{(1)b_1} , e^{(3)b_3} \rangle A(e_{(1)b_1}, e_{(2)b_2})v(e_{(3)b_3})e^{(2)b_2}$$

Does this make sense? Or have I missed something? I have read answers ( https://math.stackexchange.com/a/42108/68036 ) saying that you have to be vary of the "braiding map" and that the indices you are tracing needs to be "next to eachother" but those comments make no sense to me.

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Perhaps the braiding map is used to move the tensored vectors until they are adjacent, and then a $\langle , \rangle$ is put around them: $$tr_{(a),(b)}(T) = PutPairingAround_{(a),(b)}(MoveAdjacent_{(a),(b)}(T(e_{(1)b_1}, \dots , e_{(a) b_a} , \dots , e_{(b) b_b} , \dots , e_{(k)b_k})e^{(1)b_1}\otimes \cdots\otimes e^{(a)b_a}\otimes e^{(\tilde{a})b_{(\tilde{a})}}\otimes \cdots\otimes e^{(b)b_b}\otimes\cdots\otimes e^{(k)b_k})) =PutPairingAround_{(a),(b)}(T(e_{(1)b_1}, \dots , e_{(a) b_a} , \dots , e_{(b) b_b} , \dots , e_{(k)b_k})e^{(1)b_1}\otimes \cdots\otimes e^{(a)b_a}\otimes e^{(b)b_b}\otimes e^{(\tilde{a})b_{(\tilde{a})}}\otimes\cdots\otimes e^{(k)b_k})\\ = T(e_{(1)b_1}, \dots , e_{(a) b_a} , \dots , e_{(b) b_b} , \dots , e_{(k)b_k})\langle e^{(a)b_a},e^{(b)b_b} \rangle e^{(1)b_1}\otimes \cdots\otimes e^{(\tilde{a})b_{(\tilde{a})}}\otimes\cdots\otimes e^{(k)b_k}$$

This way extra costs/scalars can pop out while moving the tensored vectors around making it possible to be more general.

If all you want is to "shut up and calculate", you may want to look into abstract index notation, which is heavily used in those parts of physics that frequently use tensors (in particular, general relativity.) In this case, a tensor $A \in V^* \otimes V^*$ would be denoted as $$A_{ab};$$ a vector $v \in V$ would be $$v^a;$$ the tensor product $A \otimes v \in V^* \otimes V^* \otimes V$ would be $$A_{ab} v^c$$ and the tensor $A(v, -) \in V^*$, where $v \in V$, is obtained by "contracting" slots 1 & 3 of the above tensor, and is denoted as $$A_{ab} v^a.$$ The "raised" and "lowered" indices keep track of which slots belong to $V$ or $V^*$, respectively.
If you do need the different "slots" to live in different spaces (e.g., a tensor in $V_1 \otimes V_2$ where $V_1 \neq V_2$), this can be done as well, though it's not nearly as common. Usually this is done by using "primed" indices for the different vector spaces, something like $A_{a a'}$; or occasionally you'll see capitalized letters, such as $B_{aA}$. (Capital indices often tend to be reserved for spinor spaces.)