# Do 1-forms return scalars or co-vectors?

I've been browsing questions regarding 1-forms and their difference with co-vectors, and I have stumbled upon what follows.

@magma, here, said:

1-forms are simply the linear operators that take a vector and give out a number

So, thinking of a function, the range of a 1-form is a real value.

On the other hand, @Silly Goose, here, said

... if you have a 1-form α on some manifold M and you pick any x∈M, the value of α at x is a co-vector.

Doesn't the latter quote assert that a 1-form, functionally, returns a co-vector, thereby contradicting the former quote?

• A covector is an element $\omega\in T^*_xM$ and therefore defines a linear function $\omega: T_xM\rightarrow \mathbb{R}$. A $1$-form is the dual analog of a vector field. For each $x\in M$, it defines a covector $\omega(x) \in T^*_xM$. It could be called a covector field. Commented Dec 20, 2022 at 14:34

Remember that you need to put in two inputs into a 1-form before you get a scalar: a point on your manifold, and then a vector at the tangent space of that point. If you fix the point $$x\in M$$ but not a vector $$v\in T_xM$$, then you have a covector $$\omega|_x:T_xM\to \mathbb{R}$$, i.e. $$\omega|_x\in T_x^*M$$. If you fix a vector field $$X$$ on $$M$$ but not a point, then you have a function $$\omega(X):M\to \mathbb{R}$$. If you fix neither, then you have the usual definition of a 1-form, namely $$\omega:M\times T_x M\to \mathbb{R}.$$
The abstract thing that's happening is if you have a map $$X\times Y\to Z$$, you can view this same map in three ways. First, exactly as written. Second, if you fix an $$x\in X$$, this becomes a map $$Y\to Z$$. Third, if you fix a $$y\in Y$$, then this becomes a map $$X\to Z$$.
These are subtly different. All one forms on a given (assume finite dimensional vector space) are linear maps from vectors to the field. On a manifold each point has a vector space associated with it (the tangent space $$T_pM$$) upon which one can define one forms as above. The collection of all one forms is commonly denoted as $$T_pM^*$$. The difference here is that now a one form on a manifold is a function: $$\omega:M \to T_pM^*$$
And the properties must interact in a "nice" way with the smooth structure of $$M$$ or if you like the tangent bundle $$TM$$.