Linear Algebra Done Right · Problem 6A.29

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose V1,…,VmV_1, \ldots, V_m are inner product spaces. Show that the equation ⟨(u1,…,um),(v1,…,vm)⟩=⟨u1,v1⟩+⋯+⟨um,vm⟩\langle (u_1, \ldots, u_m), (v_1, \ldots, v_m) \rangle = \langle u_1, v_1 \rangle + \cdots + \langle u_m, v_m \rangle defines an inner product on V1×⋯×VmV_1 \times \cdots \times V_m. In the expression above on the right, for each k=1,…,mk = 1, \ldots, m, the inner product ⟨uk,vk⟩\langle u_k, v_k \rangle denotes the inner product on VkV_k. Each of the spaces V1,…,VmV_1, \ldots, V_m may have a different inner product, even though the same notation is used here.

Related theory

Browse the problem library