Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$.

Let $B$ be a subset of $V$ satisfying

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Linearly Independent: For all functions $f$ in $\mathbb{K}^B$, if $\displaystyle\sum_{b \in B} f(b) \cdot b = 0$, then $f$ is identically zero.

Spanning Set: For all vectors $v$ in $V$, there exists a function $f$ in $\mathbb{K}^B$ such that $\displaystyle\sum_{b \in B} f(b) \cdot b = v$.

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Let $C$ be another subset of $V$ satisfying the above conditions with $B$ replaced with $C$.

Does it follow that $|B| = |C|$?

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(I know such 'bases' don't always exist, but when they do, do they give a unique dimension?)

finitelymany terms, so we define an infinite series in terms of approximating finite series.) If you instead would like to use a sum over a subset B, please tell us exactly what the equation sum_{b in B} b = v means. – KConrad Aug 20 '10 at 2:15