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In approximation theory, a Haar space or Chebyshev space is a finite-dimensional subspace ${\displaystyle V}$ of ${\displaystyle {\mathcal {C}}(X,\mathbb {K} )}$, where ${\displaystyle X}$ is a compact space and ${\displaystyle \mathbb {K} }$ either the real numbers or the complex numbers, such that for any given ${\displaystyle f\in {\mathcal {C}}(X,\mathbb {K} )}$ there is exactly one element of ${\displaystyle V}$ that approximates ${\displaystyle f}$ "best", i.e. with minimum distance to ${\displaystyle f}$ in supremum norm.

References

1. Shapiro, Harold (1971). "2. Best uniform approximation". Topics in Approximation Theory. Lecture Notes in Mathematics. Vol. 187. Springer. pp. 19–22. doi:10.1007/BFb0058978. ISBN 3-540-05376-X.

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