- #1
Sam_Goldberg
- 46
- 1
Homework Statement
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.
Homework 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.
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.