NORMED VECTOR SPACES AND INNER PRODUCT SPACES
Normed vector spaces and inner product spaces [ edit ] Main articles: Normed vector space and Inner product space "Measuring" vectors is done by specifying a norm , a datum which measures lengths of vectors, or by an inner product , which measures angles between vectors. Norms and inner products are denoted | � | and ⟨ � , � ⟩ , respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm | � | := ⟨ � , � ⟩ . Vector spaces endowed with such data are known as normed vector spaces and inner product spaces , respectively. [39] Coordinate space � � can be equipped with the standard dot product : ⟨ � , � ⟩ = � ⋅ � = � 1 � 1 + ⋯ + � � � � . In � 2 , this reflects the common notion of the angle between two vectors � and � , by the law of cosines : � ⋅ � = cos ( ∠ ( � , � ) ) ⋅ | � | ⋅...