What Does a Linear Functional Do?

[meldraft]

Hey all,

I have been reading up on Green Functions and I stumbled upon the term "linear functional". I know the properties of the linear operator, but i can't really grasp what a functional does.

In my notes it says that it indicates a linear function whose domain is a function space, and that is maps a function to its value at a point, such that:

$$L_ξ<u>=u(ξ)</u>$$

Can someone clarify what this means? I don't understand the qualitative aspect, i.e. what it actually "does". Is it just a way to talk about the operation of going from the function domain space to its value space? What is the added value of using it instead of saying X->Y?

 

[micromass]

It's just some terminology that occurs a lot.

For example, let $\mathcal{C}(\mathbb{R},\mathbb{R})$ be the continuous functions from $\mathbb{R}$ to $\mathbb{R}$.

Then your functional is the same as saying (for example)

$$\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}:f\rightarrow f(0)$$

(where f(0) can be replaced by f(2) or f(-10) or whatever).

 

[meldraft]

Thanks for your reply!

I think I understand. I suppose though that by using the functional we can can actually discuss about the properties of the operation in a way not possible (?) by the notation you used. For instance, I am reading this in the context of generalized functions and my notes say that the functional:

$$L_g:C^0[a,b]->R$$

where g is a fixed continuous function, is not always valid. It goes on to say that there is no actual delta function $δ_ξ(x)$ such that the identity ($L^2$ inner product):

$$L_ξ<u>=<δ_ξ;u>=\int_a^bδ_ξ(x)u(x)dx=u(ξ)</u>$$

holds for every continuous function u(x), and that every (continuous) function defines a linear functional, but not conversely. To be honest I don't really understand why this is the case, but is it the use of a functional that enables us to discuss this issue?