Recent content by QuarkHead

Well, I didn't see that statement of yours. And yet... ##(V^* \otimes V)^* = V \otimes V^*## is only true in a metric space on finite-dimensional vector spaces, so that the hom-set (your RHS term ) of all Real maps from the dual space to its "parent" is equal to the tensor product of a double...

This is trivially true - ##V^*\otimes V:V \times V^* \to \mathbb{R}## is the general form, but yes, if ##V## is a vector space, then for sure so is ##V \times V^*##. Likewise, since ##\mathbb{R}## is a field it is equally a vector space. How can ##V^*\otimes V## be its own dual?

Yes, this notation is not good. As a general guidance, consider the following... Suppose ##V## an arbitrary finite-dimensional vector space. Then there will always exist a dual space ##V^*:V \to \mathbb{R}## such that for any ##\varphi \in V^*## and any ##v \in V## that ##\varphi(v) = \alpha...

Well, this is an interesting thread. Mind if I stick in my two-pennyworth? If this has been covered already, or if the discussion has moved on under its own momentum, forgive me; I am just going by the OP and a sampling of the responses. Suppose that \mathcal{C} is a category with...

After a long break doing my day job, I am back working from Fulton & Harris. So ley me pose the question Take the case of a 3-dimensional Lie algebra \mathfrak{g}. Direct computation reveals there is a basis vector, say H \in \mathfrak{g} such that, for any other X_i \in \mathfrak{g} one...

I thank you both for your responses. I was told that this is sufficient to prove uniqueness, but not existence. Maybe I was told wrong; again I thank you both

Since it appears (so far) I am infringing no rule, here is another shameless copy/paste of a thread I started on another forum, where I didn't get too much help - rather, folk tried, but confused me even further! See if you guys can do better. (Note:I am not a physicist) The mathematics here...

Let me first confess this a copy/paste of a question I asked on another forum; I trust it's not against the rules. Let M be a C^{\infty} manifold, and, for some neighbourhood U\ni p \subsetneq M let there be local coordinates x^i such that p=(x^1,\,x^2,...,x^n) Suppose that T_pM is a...