  Waring's problem, of expressing an integer as the sum of powers, has a very
long history going back to the 17th century, and the problem has been studied
in many different contexts. In this paper we introduce the notion of a Waring
subspace and a Waring identifiable subspace with respect to a projective
algebraic variety $\mathcal X$. When $\mathcal X$ is the Veronese variety,
these subspaces play a fundamental role in the theory of symmetric tensors and
are related to the Waring decomposition and Waring identifiability of symmetric
tensors (homogeneous polynomials). We give several constructions and
classification results of Waring identifiable subspaces with respect to the
Veronese variety in ${\mathbb{P}}^5({\mathbb{F}}_q)$ and in
${\mathbb{P}}^{9}({\mathbb{F}}_q)$, and include some applications to the theory
of linear systems of quadrics in ${\mathbb{P}}^3({\mathbb{F}}_q)$.
