From [C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, 2002.], page 2.
Suppose X is a real vector space, let $M \in X$ be a linear subspace, and let A $\subset$ X be nonempty. The algebraic interior of A with respect to M is
$aint_{M}A:=\{a\in X: \forall x \in M,\exists \delta>0,\forall \lambda \in[0,\delta]:a+\lambda x\in A \}.$
Is this definition worng?
Becasues it seems like that the vector $x$ should be bounded or have untity norm $\|x\|=1$.
Otherwise the condition $\forall x \in M,\exists \delta>0,\forall \lambda \in[0,\delta]:a+\lambda x\in A$ would be equivalent to the $\forall x\in M ,\|x\|=1$ the line $\{a+\lambda x:\lambda \in R\}\subset A$.
Thus $A=M$.
Clearly there is something wrong?
No, this definition is fine. How can a definition be wrong? :)
Note that $\delta$ is allowed to depend on $x$. If the norm of $x$ goes up, $\delta$ goes down.