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Uniform approximation minimizes the supremum norm of the difference between a target function and an approximant.

If continuously differentiable functions have uniformly convergent derivatives on an interval and their values converge at one point, the functions converge locally uniformly to a continuously differentiable limit whose derivative is the derivative limit. The proof integrates from the anchored point using the fundamental theorem of calculus. Uniform convergence of the functions on the entire interval follows when the interval is bounded; it need not follow on an unbounded interval.

For the free Schrodinger equation with compatible smooth initial and boundary data, subtract $g(t)e^{-x}$ before taking a Fourier sine transform. The lifted initial datum vanishes at the endpoint, making its sine transform $O(k^{-2})$ if its second derivative is integrable. The transformed forcing is $k(ig-g^{\prime })/(1+k^2)$; time integration followed by integration by parts makes its solution contribution uniformly $O(k^{-3})$. An integrable spectral majorant proves uniform convergence even at the compatible initial-boundary corner.

The supremum of pointwise distances defines a metric on maps when it is finite, for example if the target metric space is bounded. Convergence in this metric is uniform convergence. If the target is complete, Cauchy sequences have pointwise limits and the Cauchy estimate makes convergence uniform.