quantum mechanics - Distinction between state space and space of functions

In Quantum Mechanics a particle is described by its wave function $\Psi : \mathbb{R}^3\times \mathbb{R}\to \mathbb{C}$. In that sense, the state of a particle at time $t_0$ is characterized by a function $\Psi(\cdot, t_0) : \mathbb{R}^3\to \mathbb{C}$. The space of all functions like that which are suitable for a given situation is then a susbet of $L^2(\mathbb{R}^3)$ the set of square-integrable function in $\mathbb{R}^3$, equiped with the inner product

$$(\psi,\varphi) = \int_{\mathbb{R}^3}\bar{\psi}(x)\varphi(x) d^3x.$$

Now, the book I'm studying, introduces another space. The space of states $\mathcal{E}$ whose elements are kets $\left|\psi\right\rangle\in \mathcal{E}$. The author states that although isomorphic, $\mathcal{E}$ is not the space of functions I've described above.

More than that, he says that the ket $\left|\psi\right\rangle$ is the element of $\mathcal{E}$ associated to the function $\psi$.

I simply can't get the idea here of why to introduce this $\mathcal{E}$, and what $\mathcal{E}$ really is. Saying that it is a space isomorphic to the space of functions is very vague, since I believe there are tons of spaces isomorphic to it. Also, saying it is the space of kets seems vague too, because $\left|\psi\right\rangle$ as I understand is just a notation.

In that setting, why does one need to distinguish between the space of functions I've described and the space $\mathcal{E}$? In truth, what is $\mathcal{E}$ rigorously?