The inner product is a function on 🏹 Vector > Vector Space that satisfies:

1. , where the right-hand side is the conjugate of .
2. .
3. .
4. for and for .

With these properties, the inner product is linear in the second argument and conjugate-linear in the first,

A common inner product is the called the dot product,

A common 📌 Norm for vector space can also be defined via inner product as

Lastly, the inner product is also related to the 📐 Angle between vectors and as

Symmetric Positive Definite#

If 🍱 Matrix is symmetric positive definite, then

defines an inner product where and are coordinate representations with respect to basis , and .