Jtdylktuy

Recently I read this post:
can-non-linear-transformations-be-represented-as-transformation-matrices

So I concluded that normal matrices cannot represent every non-linear transformations.

But I know that some simple operators can be represented as (infinite) matrices using orthogonal function basis.

I wonder if I could represent functionals too, in vector/matrix/tensor notations.

Firstly, I want to find a way to represent multi-input functions in vector/matrix/tensor notation.

Secondly, I want to know how to represent functionals in vector/matrix/tensor notation.

For example let's say a functional $J$ is an assignment functional.

Then $$J(f;a)=f(a)$$ right?

When I turn it into vector/matrix/tensor notation:
Something like this?

$$ iota×alpha×xi $$

Where $iota$ is a vector/tensor/matrix notation of $J$, $alpha$ for $a$, and $xi$ for $f$.

Mmm, though the above's result is not a scalar as $f(a)$ should be, but something like that.

For bilinear map I previously asked a question here, so please refer if needed.

In my opinion, concatenating two operand vectors can be used to use matrix notation for binary operators. Could you verify this too?

Thank you in advance!

Edit

I think that I had heard that only linear transformations can be replaced with tensor/matrix like notations, but if you provide a proof on why even an infinite matrices/tensors (/even with infinite rank), then i'll be appreciated too.