Bilinear maps and dual spaces

Think of a function that takes two inputs and gives one output. The + operator is like that. 9+10=19 or, if you prefer to be computer-y about it, plus(9, 10) returns 19.

So is the relation “the degree to which X loves Y”. Takes as inputs two people and returns the degree to which the first loves the second. Not necessarily symmetrical! I.e. love(A→B) ≠ love(B→A). * It can get quite dramatic.

L(A,B) &notequals; L(B,A)

An operator could also take three or four inputs.  The vanilla Black-Scholes price of a call option asks for {the current price, desired exercise price, [European | American | Asian], date of expiry, volatility}.  That’s five inputs: three ⁺ numbers, one option from a set isomorphic to {1,2,3} = ℕ₃, and one date.

inputs and outputs of vanilla Black-Scholes

A bilinear map takes two inputs, and it’s linear in both terms.  Meaning if you adjust one of the inputs, the final change to the output is only a linear difference.

Multiplication is a bilinear operation (think 3×17 versus 3×18). Vectorial dot multiplication is a bilinear operation. Vectorial cross multiplication is a bilinear operation but it returns a vector instead of a scalar. Matrix multiplication is a bilinear operation which returns another matrix. And tensor multiplication , too, is bilinear.

Above, Juan Marquez shows the different bilinear operators and their duals. The point is that it’s just symbol chasing.

File:Dual Cube-Octahedron.svg

* The distinct usage “I love sandwiches” would be considered a separate mathematical operator since it takes a different kind of input.


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