Inner product space ($(V,⟨\cdot ,\cdot ⟩)$)

An inner product space is a real or complex vector space equipped with a positive-definite inner product. Over the real numbers the form is symmetric and bilinear; over the complex numbers it is conjugate symmetric and sesquilinear. It defines the norm $\Vert v\Vert =\sqrt{⟨v,v⟩}$. The Cauchy-Schwarz inequality proves the triangle inequality for this norm. A complete inner product space is a Hilbert space; every finite-dimensional inner product space is complete.