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The universal approximation property of width-bounded networks has beenstudied as a dual of the classical universal approximation theorem fordepth-bounded ones. There were several attempts to characterize the minimumwidth $w_{\min}$ enabling the universal approximation property; however, only afew of them found the exact values. In this work, we show that the minimumwidth for the universal approximation of $L^p$ functions from $[0,1]^{d_x}$ to$\mathbb R^{d_y}$ is exactly $\max\{d_x,d_y,2\}$ if an activation function isReLU-Like (e.g., ReLU, GELU, Softplus). Compared to the known result$w_{\min}=\max\{d_x+1,d_y\}$ when the domain is ${\mathbb R^{d_x}}$, our resultfirst shows that approximation on a compact domain requires smaller width thanon ${\mathbb R^{d_x}}$. We next prove a lower bound on $w_{\min}$ for uniformapproximation using general activation functions including ReLU: $w_{\min}\ged_y+1$ if $d_x<d_y\le2d_x$. Together with our first result, this shows adichotomy between $L^p$ and uniform approximations for general activationfunctions and input/output dimensions.

Minimum width for universal approximation using ReLU networks on compact domain