The class of languages that can be certified in a small amount of space 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 $X$?
(Note that X is not known to differ from NP.) ​ Yes - X is equal to PSPACE. ​ In fact, one gets that from just brute force, rather than needing Savitch's theorem.

You might also be interested in NL, which is only known to satisfy

L ​ $\subseteq$ ​ NL ​ $\subseteq$ ​ coNLOGLOGTIME-uniform SAC1 ​ ​ .

(Note that, to get a certificate characterization of NL, we probably must

"restrict the verifier to have one-way access to the certificate".)