1. Calculus In Infinite dimensional spaces

2. Integral for Banach space valued continuous maps p.1
E: Banach space
continuous map
is called the integral of
over [0,1] if
Hahn-Banach Theorem implies that y is uniquely determined if it exists and is denoted

5. Property of Integrals p.1
(ii)

7. Property of Integrals p.2
(iii) E,F : Banach spaces
continuous
Then

9. Fréchlet Derivative X,Y : normed vector spaces
f is called Fréchlet differentible at
if there is
such that
i.e

10. Remark If f is Fréchlet differentible at then in (*) is uniquely
determined. A is denoted by
is called
F-differential of f at

12. C1 map If f is Fréchlet differentible (F-differentible) at
every point of
, then f is called F-differentible on
Further,
is continuous from
into L(X,Y) with
uniform topology, then we say f is a C1 map
on Ω or

13. Chain Rule If is F-differentible at and is F- differentible at thenor

14. Example 1 If
then

16. Example 2 p.1
Let
be continuous.
Assume that
exists and continuous on

17. Example 2 p.2
Let by
Define
Then

20. Example 3
Let X be a real Hilbert space and
Define by
Then

22. Example 4 Let X be a Banach space and Then be where we identify
with T(1)

24. Exercise 1 Let X,Y be Banach spaces , be open be and with Suppose that
Show that

25. Exercise 2 Let X be a Banach spaces , be continuous and let and
Show that g is

26. VI.1 Definition. Elementary Properties Adjoint

27. Lemma VI.1 (Riesz-Lemma)
Let
For any
fixed , apply Green’s second
identity to u and
in the domain
we have
and then let