Equivalence of Definitions of Convex Set in Vector Space

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $V$ be a vector space over $\Bbb F$.

Let $C \subseteq V$.

Proof
From the definition of the linear combination $t C + \paren {1 - t} C$, we have:


 * $t C + \paren {1 - t} C = \set {t x + \paren {1 - t} y : x, y \in C}$

for $t \in \R$.

So we have:


 * $t C + \paren {1 - t} C \subseteq C$




 * $t x + \paren {1 - t} y \in C$ for all $x, y \in C$ and $t \in \closedint 0 1$.

Hence the result.