1. The problem statement, all variables and given/known data Hi guys, I'm working on a question in Tom M Apostol's second calculus book, on page 141, number 5. It is: Prove that if T (an operator on a vector space V) is linear and norm-preserving, then T is unitary. 2. Relevant equations Okay, a transformation T is linear if it satisfies (x and y are vectors, c is a complex scalar): T(x + y) = T(x) + T(y) T(cx) = cT(x) A norm-preserving transformation satisfies for all x (these are norms and inner products): |T(x)| = |x| Equivalently, (T(x), T(x)) = (x, x) Finally, T is unitary if for all x and y: (T(x), T(y)) = (x, y) Note that this is a complex vector space, so, in particular: (x , y) = (y, x)* where * is the complex conjugate. This fact is the key to my difficulties in this problem. 3. The attempt at a solution Here goes: (T(x), T(y)) = (T(x + y), T(y)) - (T(y), T(y)) = (T(x + y), T(y)) - (y , y) = (T(x + y), T(x + y)) - (T(x), T(x)) - (T(y), T(x)) - (y, y) = (x + y, x + y) - (x, x) - (y, y) - (T(y), T(x)) = (x, y) + (y, x) - (T(y), T(x)) Therefore: (T(x), T(y)) + (T(y), T(x)) = (x, y) + (y, x) Now, due to the complex conjugate rule for the inner product, I can only deduce from this equation that: (T(x), T(y)) + (T(x), T(y))* = (x, y) + (x, y)* So, I have only proven that the real parts of (T(x), T(y)) and (x, y) are equal, not the imaginary parts. I would appreciate any inputs on how to prove this too. Thanks.