ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4291 on D2043: Set of linearly independent sets
Subresult of R1406: Collection of sets ordered by inclusion contains a maximal element

Vector space always has an inclusion-maximal linearly independent set

Formulation 0

Let $R$ be a D273: Division ring.
Let $V$ be a D29: Vector space over $R$ such that

(i)	$\mathcal{L} : = \mathcal{L}_R(V)$ is the D2043: Set of linearly independent sets in $V$ over $R$
(ii)	\begin{equation} \max(\mathcal{L}) : = \{ M \in \mathcal{L} \mid \forall \, L \in \mathcal{L} : L \subseteq M \} \end{equation}

Then \begin{equation} |\max(\mathcal{L})| \geq 1 \end{equation}

Proofs

Proof 0

Let $R$ be a D273: Division ring.
Let $V$ be a D29: Vector space over $R$ such that

(i)	$\mathcal{L} : = \mathcal{L}_R(V)$ is the D2043: Set of linearly independent sets in $V$ over $R$
(ii)	\begin{equation} \max(\mathcal{L}) : = \{ M \in \mathcal{L} \mid \forall \, L \in \mathcal{L} : L \subseteq M \} \end{equation}