Another notation for the dual space

[quasar987]

My professor uses the notation Hom(V,\mathbb{R}) for the dual space of V. I usually see V* rather. What does the notation Hom(V,\mathbb{R}) stands for exactly? I suppose the domain of Hom is wider than just (vector spaces) x (their field).

The notation suggest it is the set of homomorphisms btw V and R, but this would not accurately describe V* because we want the elements of V* to be linear fct, not just group homomorphisms (we need the f(av)=af(v) part of linear too).

So does the notation comes from a more category theory perspective, where linear maps are considered as "vector space homomorphisms" or something like that?

Just thoughts. I just want to know what the general meaning for Hom is. Thx

 

[matt grime]

What is the definition of the dual space that you have that is _not_ homs into the underlying field? Hom, in the category of vector spaces _means_ linear map. It is nothing to do with group homomorphisms.

 

[quasar987]

The dual space of V to me is the vector space V* of all linear functions from V to its underlying field.

It's the first time I see the notation Hom being used for anything at all, so I am asking what it means in the most general context. Thank you.

 

[Hurkyl]

The dual space of V to me is the vector space V* of all linear functions from V to its underlying field.

It's the first time I see the notation Hom being used for anything at all, so I am asking what it means in the most general context. Thank you.

Hom(X, Y) is the object of all maps from X to Y.

The precise meaning of 'object' and 'map' depends on the context. In this case, 'object' means 'vector space' and 'map' means 'linear transformation'.

 

[Hurkyl]

Hom(X, Y) is the object of all maps from X to Y.

The precise meaning of 'object' and 'map' depends on the context. In this case, 'object' means 'vector space' and 'map' means 'linear transformation'.

I should add...

Let T : Y->W and S : X->Z be linear.
Hom(X, T) is the obvious map Hom(X, Y) -> Hom(X, W).
Hom(S, Y) is the obvious map Hom(Z, Y) -> Hom(X, Y).
Hom(S, T) is the obvious map Hom(Z, Y) -> Hom(X, W).