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This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$ certain operator (explicitly given there in terms of chosen bases) $$ f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W, $$ with $\operatorname{rank}(f\dagger g)\geqslant\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.

This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$ certain operator (explicitly given there in terms of chosen bases) $$ f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W, $$ with $\operatorname{rank}(f\dagger g)\geqslant\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.

This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$To avoid some complications, let us restrict to the case when both $V$ and $W$ are finite-dimensional.

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$ certain operator (explicitly given there in terms of chosen bases) $$ f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W, $$ with $\operatorname{rank}(f\dagger g)\geqslant\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.

This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$ certain operator (explicitly given there in terms of chosen bases) $$ f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W, $$ with $\operatorname{rank}(f\dagger g)\geqslant\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.

This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$To avoid some complications, let us restrict to the case when both $V$ and $W$ are finite-dimensional.

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$ certain operator (explicitly given there in terms of chosen bases) $$ f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W, $$ with $\operatorname{rank}(f\dagger g)=\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.

This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$To avoid some complications, let us restrict to the case when both $V$ and $W$ are finite-dimensional.

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$ certain operator (explicitly given there in terms of chosen bases) $$ f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W, $$ with $\operatorname{rank}(f\dagger g)\geqslant\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.