The converse holds also, with an additional assumption. Suppose formula_42 is a closed subspace of formula_174. If there exists a closed subspace formula_4 such that , then the projection formula_1 with range formula_42 and kernel formula_4 is continuous. This follows from the closed graph theorem. Suppose and . One needs to show that formula_205. Since formula_42 is closed and , "y" lies in formula_42, i.e. . Also, . Because formula_4 is closed and , we have formula_209, i.e. formula_210, which proves the claim.