Definition:Convex Set (Vector Space)/Definition 1
Jump to navigation
Jump to search
Definition
Let $\Bbb F \in \set {\R, \C}$.
Let $V$ be a vector space over $\Bbb F$.
Let $C \subseteq V$.
We say that $C$ is convex if and only if:
- $t x + \paren {1 - t} y \in C$
for each $x, y \in C$ and $t \in \closedint 0 1$.
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Definition $2.4$
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed Spaces
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.1$: Norms