Defining the Product of Linear Functionals: Is It Possible?

[Klaus_Hoffmann]

Let be a set of LInear functionals $$U_{n}[f]$$ n=1,2,3,4,...

so for every n $$U_{n}[\lambda f+ \mu g]=\lambda U_{n}[f]+\mu U[g]$$ (linearity)

the question is if we can define the product of 2 linear functionals so

$$U_{i}U_{j}[f]$$ makes sense.

 

[radou]

Let be a set of LInear functionals $$U_{n}[f]$$ n=1,2,3,4,...

so for every n $$U_{n}[\lambda f+ \mu g]=\lambda U_{n}[f]+\mu U[g]$$ (linearity)

the question is if we can define the product of 2 linear functionals so

$$U_{i}U_{j}[f]$$ makes sense.

You can define the product of linear functionals as composition, so you have an algebra of functionals.

 

[HallsofIvy]

Defining an "ordinary" product, that is as the product of the results of the functionals would destroy linearity.

Unfortunately, since a "functional" maps functions to numbers, the composition of two functionals does not exist.

 

[radou]

Unfortunately, since a "functional" maps functions to numbers, the composition of two functionals does not exist.

Of course, I didn't think about that.

 

[lurflurf]

One could procede as in multilinear algebra/tensor calculus and define an outer product.

Thus let u,v be functionals
u,v:V->F
(uv)f=(vf)u

One might say the product between two linear functionals is a bilinear functional.

 

[matt grime]

You can trivially define the product (as in multiplication) of two linear functionals. It just isn't a linear functional. A linear function is in particular a C/R/F valued function, so it lies in the algebra of functions, as radou sort of said.