Example · Graduate
Terms
| Term | This example | Why |
|---|
| metric space |
✓ |
The continuous functions on $[0, 1]$ with the max norm form a metric space with $d(f, g) = \norm{f - g}$ because every normed space is a metric space.
|
| normed space |
✓ |
The continuous functions 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 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.
|
| inner product space |
✗ |
The continuous functions on $[0, 1]$ with the max norm are not an inner product space, because the max norm fails the parallelogram law.
|
| hilbert space |
✗ |
The continuous functions on $[0, 1]$ with the max norm are not a Hilbert space, because max norm does not define an inner product.
|
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