180
IX – Multivariate Differential and Integral Calculus
of μ(I) shows that |p i [ν(t)] − p i [μ(t)]| is everywhere < r if ν − μ I < r
,
the differences |Dν
i (t) − Dμ
i (t)| being bounded above by ν
− μ
I . Usual
calculations then lead to the result.
Having done this, note that the notion of a differential introduced in n
◦ 2,
(i) for functions defined on an open subset of a Cartesian space generalizes
automatically to the case of a numerical-valued function F defined on an open
subset of a Banach space B or even with values in another Banach space H;
F will be said to be differentiable at a point x if there is a continuous linear
map
29 u from B to H such that, for sufficiently small h ∈ B,
F (x + h) = F (x) + u(h) + o(h)
(7.6)
with, as usual, lim o(h)/h = 0 as h tends to 0 ; then set dF (x; h) =
u(h). With this definition and taking s = 0 in formula (4’), it is tempting to
write that
dF (μ; ν) = ω [μ(1); ν(1)] − ω [μ(0); ν(0)] .
(7.7)
This expression is indeed linear in ν for given μ. To justify it, it still remains
to be shown that
F (μ + ν) = F (μ) + dF (μ; ν) + o(ν)
as the norm of the path ν ∈ C
1/2 (I, E) tends to 0. Now, (4’) and the FT
show that, if ω is closed, then
F (μ + ν) − F (μ) =
(7.8)
=
1
0
ω [μ(1) + sν(1); ν(1)] − ω
μ(0) + sν(0); ν(0)
ds ,
at least if |
|
|ν| |
| is sufficiently small so that μ + sν ∈ C
1/2 (I, G) for any s ∈ I.
But as ω(x; h) is a C
1 function of (x, h) and is linear in h, there is an equality
of the form
|ω(x + k; h) − ω(x; h)| ≤ M k.h
which holds for given x, for any h and sufficient small k. The error made by
replacing s by 0 in the functions integrated in (8) is, up to a constant factor,
bounded above by ν(1)
2 in the first case and ν(0)
2 in the second and
hence, up to a constant, the total error made is inferior to |
|
|ν| |
|
2 . But replacing
s by 0 in the integrals (8), the right hand side is replaced by expression (7).
29 as already remarked, this is a superfluous assumption. On analysis in Banach
spaces, see Dieudonn´ e, El´ ements d’Analyse, vol. 1, VIII, or Serge Lang, Analysis
I, or Henri Cartan, Calcul diff´ erentiel, etc.
IX – Multivariate Differential and Integral Calculus
of μ(I) shows that |p i [ν(t)] − p i [μ(t)]| is everywhere < r if ν − μ I < r
,
the differences |Dν
i (t) − Dμ
i (t)| being bounded above by ν
− μ
I . Usual
calculations then lead to the result.
Having done this, note that the notion of a differential introduced in n
◦ 2,
(i) for functions defined on an open subset of a Cartesian space generalizes
automatically to the case of a numerical-valued function F defined on an open
subset of a Banach space B or even with values in another Banach space H;
F will be said to be differentiable at a point x if there is a continuous linear
map
29 u from B to H such that, for sufficiently small h ∈ B,
F (x + h) = F (x) + u(h) + o(h)
(7.6)
with, as usual, lim o(h)/h = 0 as h tends to 0 ; then set dF (x; h) =
u(h). With this definition and taking s = 0 in formula (4’), it is tempting to
write that
dF (μ; ν) = ω [μ(1); ν(1)] − ω [μ(0); ν(0)] .
(7.7)
This expression is indeed linear in ν for given μ. To justify it, it still remains
to be shown that
F (μ + ν) = F (μ) + dF (μ; ν) + o(ν)
as the norm of the path ν ∈ C
1/2 (I, E) tends to 0. Now, (4’) and the FT
show that, if ω is closed, then
F (μ + ν) − F (μ) =
(7.8)
=
1
0
ω [μ(1) + sν(1); ν(1)] − ω
μ(0) + sν(0); ν(0)
ds ,
at least if |
|
|ν| |
| is sufficiently small so that μ + sν ∈ C
1/2 (I, G) for any s ∈ I.
But as ω(x; h) is a C
1 function of (x, h) and is linear in h, there is an equality
of the form
|ω(x + k; h) − ω(x; h)| ≤ M k.h
which holds for given x, for any h and sufficient small k. The error made by
replacing s by 0 in the functions integrated in (8) is, up to a constant factor,
bounded above by ν(1)
2 in the first case and ν(0)
2 in the second and
hence, up to a constant, the total error made is inferior to |
|
|ν| |
|
2 . But replacing
s by 0 in the integrals (8), the right hand side is replaced by expression (7).
29 as already remarked, this is a superfluous assumption. On analysis in Banach
spaces, see Dieudonn´ e, El´ ements d’Analyse, vol. 1, VIII, or Serge Lang, Analysis
I, or Henri Cartan, Calcul diff´ erentiel, etc.