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## 2009/11/11

### A first, crude explanation

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Im t ≅ (Im t)* ≅ V' * / Annih Im t = V' * / Ker t* ≅ Im t*

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XOO11/11/2009 6:25 pm 說：

In general, dual space often represents "adjunction" between two structures. (See p.88, MacLane's category textbook)

The isomorphism between V and V* holds only if they are finite-dimensional. It may explain that it is not completely natural to say V is isomorphic to V*, right?

Josh Ko11/13/2009 7:58 am 說：

For now I just regard V* as "transposed V", i.e., a element of V* is a row which is the transpose of an element (column) of V. This correspondence between dual spaces and row spaces does not seem to be perfect, though. For example, we have (A^T)^T = A but only V** \cong V. If you happen to have some free time, perhaps you can elaborate on "it is not completely natural to say V is isomorphic to V*"?

XOO11/18/2009 2:37 pm 說：

Given a dualization operator D, we could form an adjunction from Vec to Vec, and by the adjunction we have the unit which is a natural transformation from V to V**. However it is not a natural isomorphism in general.

Adjunction is a weaker notion of equivalence of categories, and equivalene is weaker than isomorphism of categories.

Currently, you may think adjunction as Galois connection.

Do you have a copy of MacLane's category textbook?

Josh Ko11/19/2009 10:37 am 說：

I do, but I do not have time to read it before I complete the research proposal. Hopefully I'll be able to acquaint myself with (pure) category theory next year. In fact I am now quite curious about exactly how successful category theory is (in terms of pure mathematics), and to have an answer it seems necessary to understand the theory to a certain extent. That requires much time and energy which I cannot devote to the theory now.