= Separating subspace of an algebraic dual
= Separating subspaces of an algebraic dual
{synonym}
A <linear subspace> $W\subseteq X^*$ is separating if every nonzero $x\in X$ has $T(x)\ne0$ for some $T\in W$. For finite-dimensional $X$, an evaluation map $X\to F^{\dim W}$ is <injective>, so the <rank-nullity theorem> forces $\dim W\geq\dim X=\dim X^*$ and $W=X^*$. For the infinite-dimensional <polynomial> space, the span of its coefficient functionals is a proper separating <linear subspace> of the <dual space>, which contains all sequences of coefficients, not just finitely supported ones.
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