the proof is complete. Proof for real inner products Let be a real inner product space. Consider an arbitrary pair and the function defined by Since the inner product is positive-definite, only takes non-negative values. On the other hand, can be expanded using the bilinearity of the inner product and using the fact that for real inner products: Thus, is a polynomial of degree (unless which is a case that can be independently verified). Since the sign of does not change, the discriminant of this polynomial must be non-positive: The conclusion follows. For the equality case, notice that happens if and only if If then and hence Proof for the dot product The Cauchy-Schwarz inequality in the case where the inner product is the dot product on is now proven. The Cauchy-Schwarz inequality may be rewritten as or equivalently, for which expands to: To simplify, let so that the statement that remains to be to proven can be written as which can be rearranged to The discriminant of the quadratic equation is Therefore, to complete the proof it is sufficient to prove that this quadratic either has no real roots or exactly one real root, because this will imply: Substituting the values of into gives: which is a sum of terms that are each by the trivial inequality: for all This proves the inequality and so to finish the proof, it remains to show that equality is achievable. The equality is the equality case for Cauchy-Schwarz after inspecting which proves that equality is achievable. Generalizations Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to norms. More generally, it can be interpreted as a special case of the definition of the norm of a linear operator on a Banach space (Namely, when the space is a Hilbert space). Further generalizations are in the context of operator theory, e.g. for operator-convex functions and operator algebras, where the domain and/or range are replaced by a C*-algebra or W*-algebra. An inner product can be used to define a positive linear functional. For example, given a Hilbert space being a finite measure, the standard inner product gives rise to a positive functional by Conversely, every positive linear functional on can be used to define an inner product where is the pointwise complex conjugate of In this language, the Cauchy–Schwarz inequality becomes which extends verbatim to positive functionals on C*-algebras: The next two