Can someone explain to a calculus student what "dual space is the space of linear functions" mean? I ran across this phrase today in a post and I am slightly confused. 
From my understanding, the dual space is the space of functions that sends a vector to a real number.
There are two confusions:


*

*my understanding is that a function sends a number to a real number, when I write $f(4) = 5$, I am not sending a vector to a number

*functions are already vectors that satisfies all axioms of the vector space $V$, why would be also belong in the dual space of the vector space $V^*$?

*what does "linear function" mean in this case? Why is the dual space not the space of non-linear functions such as $f(x) = sin(x)$
Sorry if my questions seem silly!
 A: Fix a vector space $V$, finite-dimensional, say. The dual space is defined by $$V^\ast = \{ f: V \to \Bbb R \mid f \text{ is linear}   \}.$$

my understanding is that a function sends a number to a real number, when I write $f(4)=5$, I am not sending a vector to a number

If $f: \Bbb R \to \Bbb R$ is linear, and you're looking the first $\Bbb R$ as the vector field, and the second $\Bbb R$ as the underlying field, then $f \in \Bbb R^\ast$. You're applying $f$ in a vector: $4$ (an element of the first $\Bbb R$).

functions are already vectors that satisfies all axioms of the vector space $V$, why would be also belong in the dual space of the vector space $V^∗$?

Vectors are elements of a vector space. So functions are vectors, because they're elements of a vector space (the dual space of the initial space). Functions don't necessarily verify the axioms for $V$. They will do so for $V^\ast$, with the operations defined pointwise.

what does "linear function" mean in this case? Why is the dual space not the space of non-linear functions such as $f(x)=\sin(x)$

Given $V,W$ vector spaces over the same field, say, $\Bbb R$, we say that $T: V \to W$ is linear if $T(x+\lambda y) = T(x)+\lambda T(y)$ for all $x,y \in V$ and for all $\lambda \in \Bbb R$. The dual space only consider the linear functions by definition, since they're have special properties and applications (such as linear approximations, derivatives, etc), and are easier to work with than arbitrary functions.
