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In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we refer to as approximation spaces of artificial neural networks and we present several tools to handle those spaces. Roughly speaking, approximation spaces consist of sequences of functions which can, in a suitable way, be approximated by ANNs without curse of dimensionality in the sense that the number of required ANN parameters to approximate a function of the sequence with an accuracy $\varepsilon > 0$ grows at most polynomially both in the reciprocal $1/\varepsilon$ of the required accuracy and in the dimension $d \in \mathbb{N} = \{1, 2, 3, \ldots \}$ of the function. We show that these approximation spaces are closed under various operations including linear combinations, formations of limits, and infinite compositions. To illustrate the utility of the machinery proposed in this paper, we employ the developed theory to prove that ANNs have the capacity to overcome the curse of dimensionality in the numerical approximation of certain first order transport partial differential equations (PDEs). We even prove that approximation spaces are closed under flows of first order transport PDEs.