Example · Graduate
Terms
| Term | This example | Why |
|---|
| metric space |
✓ |
The polynomials on $[0, 1]$ with the max norm form a metric space, because every normed space is a metric space, with $d(p, q) = \norm{p - q}$.
|
| normed space |
✓ |
The polynomials on $[0, 1]$ with the max norm form a normed space, because the max norm satisfies the triangle inequality and scales correctly.
|
| Banach space |
✗ |
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 $e^x$.
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