Linearly Independent Set is Contained in some Basis

Theorem

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Finite Dimensional Case

Let $E$ be a vector space of $n$ dimensions.

Let $H$ be a linearly independent subset of $E$.

There exists a basis $B$ for $E$ such that $H \subseteq B$.

Infinite Dimensional Case

Let $K$ be a field.

Let $E$ be a vector space over $K$.

Let $H$ be a linearly independent subset of $E$.

There exists a basis $B$ for $E$ such that $H \subseteq B$.