  Let $f$ be a polynomial in the free algebra over a field $K$, and let $A$ be
a $K$-algebra. We denote by $\S_A(f)$, $\A_A(f)$ and $\I_A(f)$, respectively,
the `verbal' subspace, subalgebra, and ideal, in $A$, generated by the set of
all $f$-values in $A$. We begin by studying the following problem: if $\S_A(f)$
is finite-dimensional, is it true that $\A_A(f)$ and $\I_A(f)$ are also
finite-dimensional? We then consider the dual to this problem for `marginal'
subspaces that are finite-codimensional in $A$. If $f$ is multilinear, the
marginal subspace, $\widehat{\S}_A(f)$, of $f$ in $A$ is the set of all
elements $z$ in $A$ such that $f$ evaluates to 0 whenever any of the
indeterminates in $f$ is evaluated to $z$. We conclude by discussing the
relationship between the finite-dimensionality of $\S_A(f)$ and the
finite-codimensionality of $\widehat{\S}_A(f)$.
