In a variational autoencoder (VAE), we have an encoder network $E_{\phi}$ that maps inputs $x$ to the distribution parameters of the approximate posterior $q_{\phi}(z \vert x)$. Most commonly we model this distribution over latent as a diagonal-covariance Gaussian, so we have
$$ (\mu_{\phi}(x), \Sigma_{\phi}(x)) = E_{\phi}(x) $$
Where $\Sigma_{\phi}(x)$ are the diagonal elements of the covariance matrix corresponding to datapoint $x$. This results in a different set of distribution parameters for each datapoint $x$, where the computational cost of learning the parameters of the posterior are 'amortized' through learning the model parameters of the encoder $E_{\phi}$.
In view of this, can the distribution over the latent space produced by the VAE be thought of as a very large mixture of Gaussians with a number of components equal to the number of data points we have? i.e. for $N$ data points $\{x_1, x_2, \ldots x_N\}$, do we have
$$ q_{\phi}(z \vert x) = \frac{1}{N} \sum_{n=1}^N \delta(x,x_n) \mathcal{N}\left(\mu_{\phi}(x_n), \Sigma_{\phi}(x_n)\right)$$
In other words, can the latent variable $Z$ be modelled as a mixture of $N$ equally weighted Gaussian components, with component $n$ having distribution $\mathcal{N}\left(\mu_{\phi}(x_n), \Sigma_{\phi}(x_n)\right)$?