There seem to be two main classes of duals.

One from linear algebra / functional analysis, and one from optimization theory. (Lagrangian/convex duals)

Well actually there is.

Recall that dual space of a vector space V is simply the set of all linear functionals of V. To put it another way, dual space is simply all the row vectors if the original space is finite dimensional.

Dual norm comes up in convex optimization. Dual norm of z is defined as sup_x {

This really plays a role in Lp duality. Lp dualtity says that the dual of Lp space is isometric to Lq. Here we are talking about infinite-dimensional spaces. Also the dual norm of an element of the dual space of Lp will be equal to the Lq norm of that linear functional when mapped into Lq. (See any real analysis text)

What about the dual cone?

You can think about the dual cone as one way of generating a INDUCED CONE IN THE DUAL SPACE. (Again, we think of row vectors as elements in the dual space)

I don't see the relation between the Fenchel dual (Legendre transform), Lagrangian dual, and the dual space though.

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