if which happens if and only if for all scalars The orthogonal complement of a subset is which is always a closed vector subspace of The Hilbert projection theorem guarantees that for any nonempty closed convex subset of a Hilbert space there exists a unique vector such that that is, is the (unique) global minimum point of the function defined by Statement Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references). Let denote the underlying scalar field of Fix Define by which is a linear functional on since is in the linear argument. By the Cauchy–Schwarz inequality, which shows that is bounded (equivalently, continuous) and that It remains to show that By using in place of it follows that (the equality holds because is real and non-negative). Thus that The proof above did not use the fact that is complete, which shows that the formula for the norm holds more generally for all inner product spaces. Suppose are such that and for all Then which shows that is the constant linear functional. Consequently which implies that Let If (or equivalently, if ) then taking completes the proof so assume that and The continuity of implies that is a closed subspace of (because and is a closed subset of ). Let denote the orthogonal complement of in Because is closed and is a Hilbert space, can be written as the direct sum (a proof of this is given in the article on the Hilbert projection theorem). Because there exists some non-zero For any which shows that where now implies Solving for shows that which proves that the vector satisfies Applying the norm formula that was proved above with shows that Also, the vector has norm and satisfies It can now be deduced that is -dimensional when Let be any non-zero vector. Replacing with in the proof above shows that the vector satisfies for every The uniqueness of the (non-zero) vector representing implies that which in turn implies that and Thus every vector in is a scalar multiple of The formulas for the inner products follow from the polarization identity. Observations If then So in particular, is always real and furthermore, if and only if if and only if Linear functionals as affine hyperplanes A non-trivial continuous linear functional is often interpreted geometrically by identifying it with the affine hyperplane (the kernel is also often visualized