Inner-Product-Induced Metric Space

analysis topology

Definition

Inner-Product-Induced Metric Space

An inner-product-induced metric space is the metric space obtained from an inner product space by using its inner product to define distance.

Construction

Let $⟨V,⟨\cdot ,\cdot ⟩⟩$ be an inner product space. The inner product first induces the norm

$$\Vert x\Vert =\sqrt{⟨x,x⟩}.$$

This norm then induces the metric

$$d(x,y)=\Vert x-y\Vert =\sqrt{⟨x-y,x-y⟩}.$$

Thus the induced metric space is $⟨V,d⟩$.