Re:The Geometric Tool That Solved Einstein's Relativity Problem
by senderista on Monday August 12, 2024 @03:09PM
(#41228167)
I think you're describing the evaluation map T(v, w) = w(v), which has type (1,1), rather than the inner product, which has type (2,0). The inner product lets you "raise and lower indices" (i.e. convert between vectors and dual vectors), so you can basically pretend that it is the evaluation map.
Re:The Geometric Tool That Solved Einstein's Relativity Problem
by will-burner on Tuesday August 13, 2024 @11:42AM
(#41236532)
The dot product! That's a good example, thank you.
It's multilinear because it's linear in each of its arguments separately: <ca, b> = c<a,b> and <a, cb> = c<a,b>.
Another simple but less obvious example is a rotation (orthogonal) matrix. It takes a vector as an input, and returns a vector. But a vector itself can be thought of as a linear function that takes a dual vector and returns a number (via the inner product, above!). So, applying the rotation matrix to a vector is a sort of "currying" on the multilinear map, while the matrix alone can be considered a function that takes a vector and a dual vector, and returns a number.
In functional notation, you can consider your rotation matrix to be a function (V x V*) -> K, which can in turn be considered a function V -> (V* -> K), where V* is the dual space of V.