Let $G_m(\mathbb R^m) := \left\{ g: \mathbb R^m \rightarrow \mathbb R, g(x) = \langle w_2, \phi(W_1 x + b) \rangle \ \vert \ n < \infty, W_1\in \mathbb R^{n\times m}, w_2, b\in \mathbb R^n\right\}$. Also, the function $\phi$ is given componentwise by $\phi(z) = \left(\phi^{\star}\left(z_i\right)\right)_{i = 1, \dots, n}$ for any $n\in \mathbb R^n$, where $\phi^{\star}: \mathbb R\rightarrow \mathbb R$ is bounded and nonconstant.
Now, let us consider the space $L^{p}(\mathbb R^m)$, $1\leq p < \infty$ for a finite measure $\mu$ on $\mathbb{R}^m$.
My question would be: My teacher said that $G_m(\mathbb R^m)$ is a linear subspace of $L^p(\mathbb R^m)$, but I have problems proving that $G_m(\mathbb R^m)$ is a subspace of $L^p(\mathbb R^m)$. So let $g\in G_m(\mathbb R^m)$, then we need to show that $\vert\vert g\vert\vert_p = \left( \int_{\mathbb R^m} \vert g\vert^p d\mu \right)^{1/p} = \left( \int_{\mathbb R^m} \vert \langle w_2, \phi(W_1 x + b)\rangle\vert^p d\mu \right)^{1/p} < \infty$. But how do I prove this?
(The context of this question is Hornik's famous $L^p$ universal approximation result for a feedforward neural network.)