Nonlinear Equations - UFRJ

[SEC. 5.5: COMPACTIFICATIONS 71
Exercise 5.3. The Frobenius norm for tensors T i1···ip
j 1···j q
n∑
‖T ‖ F = √ |T i1···ip
j 1···j q
| 2
i 1,··· ,j q=1
The unitary group acts on the variable j 1 by composition:
is
T i1···ip
j 1···j q
U
N
∑
k=1
T i1···ip
k···j q
U k j 1
.
Show that the Frobenius norm is invariant for the U(n)-action. Deduce
that it is invariant when U(n) acts simultaneously on all lower
(or upper) indices. Deduce that Weyl’s norm is invariant by unitary
action f f ◦ U.
Exercise 5.4. This is another proof that the inner product defined
in (5.2) is U(n + 1)-invariant. Show that for all f ∈ H d ,
‖f‖ 2 = 1 ∫
2 d ‖f(x)‖ 2 1
/2
d! C (2π) n+1 dV (x).
e−‖x‖2
n+1
The integral is the L 2 norm of f with respect to zero average, unit
variance probability measure. Conclude that ‖f‖ is invariant.
Exercise 5.5. Show that if F = H d , then the induced norm defined
in Lemma 5.10 is d times the Fubini-Study metric. Hint: assume
without loss of generality that x = e 0 .