NP can be characterized in two different ways, one of them is that it's the class of languages that can be certified by a witness in a polynomial time. I wonder, if we consider the same notion but with space considerations to get a class $X$, Is such a class known to be of any importance? Why? To be more precise, $L\in X$ iff there exists a TM $M$ s.t. $\forall x\in \{0,1\}^* (x\in L \iff  \exists u \in \{0,1\}^{p(n)}$ s.t. $M(x,u)=1$ s.t. $M$ uses space bounded by a polynomial $P$.Clearly, NP$\subset$ X. Is there any interesting thing that we can know about $X$?
The class of languages that can be certified in a small amount of space
complexity theory;complexity classes;space complexity