Are functionals and operators the same thing?

[gentsagree]

Are functionals a special case of operators (as written on Wiki)?

Operators are mappings between two vector spaces, whilst a functional is a map from a vector space (the space of functions, say) to a field [or from a module to a ring, I guess]. Now, the field is NOT NECESSARILY a vector space. It could be the field over which the vector space is defined.

Can somebody clear this up for me?

arigato.

 

[economicsnerd]

[Whenever we talk about vector spaces, there's some underlying field $\mathbb F$ in the background, and all the vector spaces under consideration are over the same field $\mathbb F$.]

There's a natural way of viewing $\mathbb F$ itself as a vector space, with addition being field addition and scalar multiplication being field multiplication. Viewing $\mathbb F$ as a vector space in this way, a linear functional on the vector space $V$ is just a linear transformation $V\to\mathbb F$.

Depending who you ask, a linear operator could either mean an arbitrary linear transformation $V\to W$ (in which case a linear functional is indeed a special case with $W=\mathbb F$), or it's the special case of a linear transformation $V\to V$ (in which case it's distinct from a linear functional).

 

[R136a1]

A field is always a vector space over itself (or can be seen as such). So $\mathbb{R}$ is canonically an $\mathbb{R}$-vector space. In that sense, a functional is always a special operator.

Note however that some authors tend to use operator and functional in a completely different way.

 

[gentsagree]

Thank you for the good replies. One more question pops to mind:

What do we mean exactly when we say that we declass an operator to a function (say, in path integrals)?

 

[blue_raver22]

Functionals and operators are not the same thing. While both involve mappings between vector spaces, functionals specifically map from a vector space to a field or ring, whereas operators map between two vector spaces. Functionals can be seen as a special case of operators, where the second vector space is the field or ring in which the functional is defined. However, not all operators can be considered as functionals. It is important to understand the specific definitions and properties of each concept in order to use them correctly in scientific contexts.

 

[blue_raver22]

Functionals and operators are not the same thing. While both are mathematical objects that map one set of elements to another, they differ in the types of spaces they operate on. Operators map between two vector spaces, while functionals map from a vector space to a field or ring.

In the context of linear algebra, operators are typically used to represent linear transformations between vector spaces, while functionals are used to represent linear maps from a vector space to its underlying field. This means that functionals are a special case of operators, as they are a specific type of mapping that operates on a specific type of space.

However, it is important to note that the field or ring that a functional maps to does not necessarily have to be a vector space. This is where the distinction between functionals and operators becomes clearer. While an operator must always map between two vector spaces, a functional can map to any type of field or ring.

In summary, functionals and operators are related but not the same. Functionals are a specific type of operator that map from a vector space to a field or ring, while operators are more general mappings between two vector spaces.