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Results tagged with banach-spaces
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user 4354
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
14
votes
Accepted
Point on a line nearest a point in Banach space
The answer is no in dimension 2 and yes in dimension 3 and higher. The property that the nearest-point projection to a line does not increase the norm is equivalent to the symmetry of orthogonality re …
- 32.4k
6
votes
Accepted
Inequalities for uniformly convex normed spaces
If the second $\delta(\varepsilon)$ is allowed to differ from the first one, then there is a simple implicit argument: Suppose the contrary, then there is a sequence $X_n$ of 2-dimensional normed spac …
- 32.4k
2
votes
Shape of long sequences in C(ω_1)
I can partially answer the second question. If $X$ is a compact Hausdorff space whose topology has a countable base at every point [Edit: $\omega_1$ has this property but not compact], then there are …
- 32.4k
4
votes
Accepted
Analogue of an orthogonal subspace in a noneuclidian normed space
Here is a simple proof that the property holds only for Euclidean norms, at least if the norm in question is $C^1$ smooth and strictly convex. Surely it was known way before Gromov was born.
Let $S$ …
- 32.4k