Linear Algebra Done Right · Problem 9D.13

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · tensor-products

Problem

Suppose that

V_1, \ldots, V_m

are finite-dimensional inner product spaces and

V_1 \otimes \cdots \otimes V_m

is made into an inner product space using the inner product from Exercise 12. Suppose

e^k_1, \ldots, e^k_{n_k}

is an orthonormal basis of

V_k

for each

k = 1, \ldots, m

. Show that the list

\{e^1_{j_1} \otimes \cdots \otimes e^m_{j_m}\}_{j_1=1,\ldots,n_1;\cdots;j_m=1,\ldots,n_m}

is an orthonormal basis of

V_1 \otimes \cdots \otimes V_m