Singleton is Convex Set

Theorem

Let $V$ be a vector space over $\R$ or $\C$, and let $v \in V$.

Then the singleton $S = \set v$ is a convex set.

Proof

For any $x, y \in S$, we have $x = y = v$.

It follows that:

$\forall t \in \closedint 0 1: t x + \paren {1 - t} y = v \in S$

Hence $S$ is a convex set.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality