16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 11

Inner Product Spaces

Mathematics — Linear Algebra · inner-product-on-operators

Problem

Let

V

be an

n

-dimensional inner product space over the field

F

, and let

L(V,V)

be the space of linear operators on

V

. Show that there is a unique inner product on

L(V,V)

with the property that

||T_{\alpha,\beta}||^2 = ||\alpha||^2||\beta||^2

for all

\alpha, \beta

V

. (

T_{\alpha,\beta}

is the operator defined in Exercise 10.) Find an isomorphism between

L(V,V)

with this inner product and the space of

n \times n

matrices over

F

, with the inner product

(A|B) = \text{tr}(AB^*)