Inner products

• How Do You Expand an Inner Product of Finite Sums?
  Expand an inner product of finite sums by bilinearity or sesquilinearity, then simplify cross terms carefully, especially when calculating a squared norm.
• What Is an Example of a Nonstandard Inner Product?
  A weighted dot product such as 2x₁y₁ + 3x₂y₂ is a valid inner product that differs from the standard one in the same coordinates.