An inner product turns vectors into linear measurements. In a Euclidean vector space, each vector $\mathbf{p}$ defines a covector

$$\alpha_{\mathbf{p}}(\mathbf{v})=\mathbf{p}\cdot\mathbf{v}$$

In finite-dimensional Euclidean space, every linear functional can be represented uniquely in this form once the inner product is fixed. This is why dot products can be viewed not only as geometric comparisons between two vectors, but also as scalar-valued linear maps.

The identification depends on the inner product. Vectors and covectors remain conceptually distinct objects even when Euclidean geometry gives a natural correspondence between them.

Visual intuition: dot products and duality