Let $\mathbf{Hilb}$ be the category of Hilbert spaces and continuous linear maps. Turn it into a symmetric monoidal category using the tensor product of Hilbert spaces. What are the dualizable objects?
2026-03-25 12:40:12.1774442412
What are the dualizable objects in the category of Hilbert spaces?
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It is well-known that finite-dimensional vector spaces are precisely the dualizable objects in $(\mathbf{Vect}_K,\otimes)$. The same holds here: Finite-dimensional Hilbert spaces are precisely the dualizable objects in $(\mathbf{Hilb},\otimes)$. This is mentioned here without proof, but I assume that you can almost recycle the argument from vector spaces.