Nonlinear Equations - UFRJ

[SEC. 7.1: THE GAMMA INVARIANT 85
Before proving the only if part of Proposition 7.2, we need to relate
the norm of a multi-linear map to the norm of the corresponding
polynomial.
Lemma 7.3. Let k ≥ 2. Let T : E k → F be k-linear and symmetric.
Let S : E → F, S(x) = T (x, x, . . . , x) be the corresponding
polynomial. Then,
‖T‖ ≤ e k−1
sup
‖x‖≤1
‖S(x)‖
Proof. The polarization formula for (real or complex) tensors is
(
T(x 1 , · · · , x k ) = 1 ∑
k∑
)
2 k ɛ 1 · · · ɛ k S ɛ l x l
k!
l=1
ɛ j=±1
j=1,...,k
It is easily derived by expanding the expression inside parentheses.
There will be 2 k k! terms of the form
ɛ 1 · · · ɛ k T (x 1 , x 2 , · · · , x k )
or its permutations. All other terms miss at least one variable (say
x j ). They cancel by summing for ɛ j = ±1.
It follows that when ‖x‖ ≤ 1,
( k∑
)
T(x 1 , · · · , x k ) ≤ 1 max ‖S ɛ l x l ‖
k! ɛ j=±1
j=1,...,k l=1
≤
kk
k!
sup ‖S(x)‖
‖x‖≤1
The Lemma follows from using Stirling’s formula,
We obtain:
‖T‖ ≤
k! ≥ √ 2πkk k e −k e 1/(12k+1) .
( ) 1
√ e 12k+1 e k sup ‖S(x)‖.
2πk ‖x‖≤1
Then we use the fact that k ≥ 2, hence √ 2πk ≥ e.