R^n/Standard inner product/Example

On ${}\mathbb {R} ^{n}$, the mapping

$\mathbb {R} ^{n}\times \mathbb {R} ^{n}\longrightarrow \mathbb {R} ,(v,w)=({\left(v_{1},\ldots ,v_{n}\right)},{\left(w_{1},\ldots ,w_{n}\right)})\longmapsto \sum _{i=1}^{n}v_{i}w_{i},$

defines an inner product, which is called the standard inner product. Simple computations show that this is indeed an inner product.