I'm reading through Steinbach's book on Eliptic Boundary Value Problems and struggling with understanding of the definition of Sobolev space
He defines it as $$H^s(\mathbb R^n) := \{u \in \mathcal S^*(\mathbb R^n) \colon \mathcal J^s u \in L_{2}(\mathbb R^n) \}$$
Where $S(\mathbb R^n)$ is Schwartz space, $\mathcal J^s$ is a Bessel Operator and $$S^*(\mathbb R^n):= \{T:S(\mathbb R^n) \mapsto \mathbb C : \text{ T linear functional}\}$$
But it can be also defined as: $$H^s(\mathbb R^n) := W_{2}^{s}(\mathbb R^n) := \{u \in L_{2}(\mathbb R^n) : D^{a}u \in L_{2}(\mathbb R^n),|a| \leq s \} $$
I am struggling to understand how these definitions are the same, since from the first definition each element is a complex functional from Schwartz Space to C, whereas from the second definition each element is an integrable measurable function with integrable measurable weak derivatives.
Thank you.