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. (The context of this question is Hornik's famous $L^p$ universal approximation result for a feedforward neural network.)
I already convinced myself here that $G_m(\mathbb R^m)$ is a subset of $L^p(\mathbb R^m)$, $1\leq p< \infty$. What I would now like to show is that $G_m(\mathbb R^m)$ is also a subspace of $L^p(\mathbb R^m)$. So:
- $0 \in G_m(\mathbb R^m)$,
- $g\in G_m(\mathbb R^m) \Rightarrow \lambda g\in \mathbb R^m, \lambda\in\mathbb R$.
The only thing missing is showing that if $g_1$, $g_2 \in \mathbb{R}^m$, then also $g_1 - g_2 \in \mathbb R^m$ (i.e., that our subset is closed under subtraction). Let $g_1(x) = \left\langle \tilde{w}_2, \phi\left( \tilde{W}_1 x + \tilde{b} \right)\right\rangle$, $g_2\left(x\right) = \left\langle w_2, \phi\left(W_1 x + b\right)\right\rangle$.
Unfortunately, I cannot use the property $\left\langle u + v, w\right\rangle = \langle u, w\rangle + \langle v, w\rangle$, so I am not sure how to proceed. However, our teacher told us that $G_m(\mathbb R^m)$ is a subspace of $L^p$.
Thank you very much in advance!