Banach space
Graduate
A Banach space is a normed space that is complete: every Cauchy sequence converges.
Examples
Examples
- $\reals^n$ with the Euclidean norm Undergraduate
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Continuous functions on $[0, 1]$ with the max norm
Graduate
The continuous functions on $[0, 1]$ with the max norm form a Banach space, because a uniform limit of continuous functions is continuous, so the space is complete.
Non-examples
Things that may be mistaken for examples.
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Polynomials on $[0, 1]$ with the max norm
Graduate
The polynomials on $[0, 1]$ with the max norm are not a Banach space, because, by the Weierstrass approximation theorem, polynomials converge uniformly to non-polynomial functions such as $e^x$.
Properties
- Every finite-dimensional normed space is a Banach space.