Definition:Uniformly Convex Normed Vector Space
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Definition
Let $\struct {X, \norm \cdot}$ be a normed vector space.
We say that $X$ is uniformly convex if and only if:
- for every $\epsilon > 0$ there exists $\delta > 0$ such that:
- whenever $x, y \in X$ have $\norm x = \norm y = 1$ and $\norm {x - y} > \epsilon$, we have:
- $\ds \norm {\frac {x + y} 2} < 1 - \delta$
- whenever $x, y \in X$ have $\norm x = \norm y = 1$ and $\norm {x - y} > \epsilon$, we have:
Also see
- Results about uniformly convex normed vector spaces can be found here.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $8$: Exercises