We study the space complexity of the cutting planes proof system, in which the lines in a proof are integral linear inequalities. We measure the space used by a refutation as the number of inequalities that need to be kept on a blackboard while verifying it. We show that any unsatisfiable set of inequalities has a cutting planes refutation in space five. This is in contrast to the weaker resolution proof system, for which the analogous space measure has been well-studied and many optimal lower bounds are known.

Motivated by this result we consider a natural restriction of cutting planes, in which all coefficients have size bounded by a constant. We show that there is a CNF which requires super-constant space to refute in this system. The system nevertheless already has an exponential speed-up over resolution with respect to size, and we additionally show that it is
stronger than resolution with respect to space, by constructing constant-space cutting planes proofs of the pigeonhole principle with coefficients bounded by two.

We also consider variable space for cutting planes, where we count the number of instances of variables on the blackboard, and total space, where we count the total number of symbols.