Understanding Distributions I have been studying distributions, and I am still trying to get the intuition behind the following:
$1)$ Suppose $f\in L^1_{loc}(\mathbb{R})$, and $\phi\in C^{\infty}_c(\mathbb{R})$. A distribution is defined as $\langle T_f, \phi \rangle=\int_{\mathbb{R}}f\phi dx$. 
Suppose we are given an $f$, is the definition telling us we can define a distribution for any $\phi$? In other words, if we have some given $L^1_{loc}$ function, we can define a different distribution for say $\phi_1=sin(x)\chi_{[0,1]}$ and $\phi_2=e^x\chi_{[0,2]}$? 
$2)$ From this definition, is it valid to define a function as follows:
$g(x)=\int_{\mathbb{R}}f\phi dx$ with $x\in [x-\epsilon, x+\epsilon]$ and $\phi$ compactly supported on $[x-\epsilon, x+\epsilon]$? where $f$ and $\phi$ are some appropriate functions?
$3)$ If $S(\mathbb{R})$ is the Schwarz space of rapidly decreasing functions, and $S'(\mathbb{R})$ is the dual space (space of tempered distributions), does this just mean that for $f\in S(\mathbb{R})$, the Fourier transform $\hat f$ defines a distribution which takes finite values? 
I think I am most stuck trying to make sense of the third one. 

For 1) there is not a distribution for every $\phi$. The distribution is the linear functional
$$\phi \longmapsto \int_{\mathbb R} f \phi dx$$
This functional is denoted $T_f$.
