linear algebra - Uniqueness of dimension for topological vector spaces

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 $displaystylesum_{b in B} f(b) cdot b = 0$, then $f$ is identically zero.

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

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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?)

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