The definition of generalized function in my books is as follows: generalized function is linear continuous function on fundamental spaces, such as $\mathbb{D}(R^n),\varphi(R^n)$.
But in ADAMS, the definition of generalized function means: $\mathbb{D}'(R^n)$ (which is the linear continuous function on $\mathbb{D}(R^n)$, where $\mathbb{D}(R^n)$ means $C_c^\infty(R^n)$ )
so what is the exact definition of generalized function? I'm so confused of it. Somebody can help me to make it clear? Thanks for your attention.
If $X$ is a topological vector space, the notation $X'$ is used to represent its dual, that is, the space of continuous linear functionals on $X$. Then, $\Bbb D'(\Bbb R^n)$ is the space of linear continuous functions on $\Bbb D(\Bbb R^n)$. Both definitions coincide.