The above argument makes use of the assumption that both formula_42 and formula_4 are closed. In general, given a closed subspace formula_42, there need not exist a complementary closed subspace formula_4, although for Hilbert spaces this can always be done by taking the orthogonal complement. For Banach spaces, a one-dimensional subspace always has a closed complementary subspace. This is an immediate consequence of Hahn–Banach theorem. Let formula_42 be the linear span of formula_116. By Hahn–Banach, there exists a bounded linear functional formula_217 such that . The operator formula_218 satisfies formula_2, i.e. it is a projection. Boundedness of formula_217 implies continuity of formula_1 and therefore formula_222 is a closed complementary subspace of formula_42.