Inner Product and Normed Vector Spaces

Vector spaces may have additional structure. For instance, there may be a natural notion of length of a vector and/or angle between vectors. The properties of length and angle will be formally captured in the notions of norm and inner product. These notions require us to restrict ourselves to vector spaces over R or C. Indeed, a length is always nonnegative and thus we will need the inequalities ≤,≥, <,> (with properties like x, y ≥ 0 ⇒ xy ≥ 0 and x ≥ y ⇒ x+ z ≥ y + z).