On the finite basis property of abstract $T$-spaces

Let $F=k\langle x_1,\dots,x_i,\dots\rangle$ be the free countably generated algebra over a field $k$ of the characteristic 0. A vector subspace $V$ of the algebra $F$ is called a $T$-space of $F$ if it is closed under substitutions. It is clear that an ideal $I$ of $F$ is a $T$-ideal if and only if $I$ is a $T$-space of $F$. The aim of this paper is to introduce the definition of the abstract $T$-space and to prove the finite basis property for some large class of $T$-spaces.
The main result of this paper is the following
Theorem. Let $I$ be a $T$-ideal of algebra $F$ which contains a Capelly polynomial. Then every $T$-space of $F/I$ is finitely based.
The statement of this theorem allows us to give a positive answer to the local Specht's problem (A. Kemer gave a positive answer to Specht's problem using another approach) and to the representability problem.