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1
f (x + v) ’ f (x) = f (x + t v)(v) dt,
0
and hence
1
f (x + s v) ’ f (x)
’ f (x)(v) = f (x + t s v) ’ f (x) (v) dt ’ 0,
s 0
which converges to 0 for s ’ 0 uniformly for v in any bounded set, since f (x +
t s v) ’ f (x) uniformly on bounded sets for s ’ 0 and uniformly for t ∈ [0, 1] and
v in any bounded set, since f is assumed to be continuous.
Recall furthermore that a mapping f : E ⊇ U ’ F on a Banach space E is called
Lipschitz if
f (x1 ) ’ f (x2 )
: x1 , x2 ∈ U, x1 = x2 is bounded in F.
x1 ’ x2
It is called H¨lder of order 0 < p ¤ 1 if
o
f (x1 ) ’ f (x2 )
: x1 , x2 ∈ U, x1 = x2 is bounded in F.
x1 ’ x2 p


13.3
13.4 13. Di¬erentiability of seminorms 129

13.4. Lemma. Gˆteaux-di¬erentiability of convex functions. Every convex
a
function q : E ’ R has one sided directional derivatives. The derivative q (x) is
sublinear and locally bounded (or continuous) if q is locally bounded (or continuous).
In particular, such a function is Gˆteaux-di¬erentiable at x if and only if q (x) is
a
an odd function, i.e. q (x)(’v) = ’q (x)(v).

If E is not normed, then locally bounded-ness should mean bounded on bornologi-
cally compact sets.

Proof. For 0 < t < t we have by convexity that
t t t t
q(x + t v) = q (1 ’ )x + (x + t v) ¤ (1 ’ ) q(x) + q(x + t v).
t t t t

Hence q(x+t v)’q(x) ¤ q(x+t tv)’q(x) . Thus, the di¬erence quotient is monotone
t
falling for t ’ 0. It is also bounded from below, since for t < 0 < t we have
t t
(x + t v) + (1 ’
q(x) = q ) (x + t v)
t’t t’t
t t
¤ q(x + t v) + (1 ’ ) q(x + t v),
t’t t’t
q(x+t v)’q(x) q(x+t v)’q(x)
¤
and hence . Thus, the one sided derivative
t t

q(x + t v) ’ q(x)
lim
t
t 0

exists.
As a derivative q (x) automatically satis¬es q (x)(t v) = t q (x)(v) for all t ≥ 0. The
derivative q (x) is convex as limit of the convex functions v ’ q(x+tv)’q(x) . Hence
t
it is sublinear.
The convexity of q implies that

q(x) ’ q(x ’ v) ¤ q (x)(v) ¤ q(x + v) ’ q(x).

Therefore, the local boundedness of q at x implies that of q (x) at 0. Let := f (x),
then subadditivity and odd-ness implies (a) ¤ (a + b) + (’b) = (a + b) ’ (b)
and hence the converse triangle inequality.

Remark. If q is a seminorm, then q(x+tv)’q(x) ¤ q(x)+t q(v)’q(x)
= q(v), hence
t t
q(x+t x)’q(x)
q (x)(v) ¤ q(v), and furthermore q (x)(x) = limt 0 = limt 0 q(x) =
t
q(x). Hence we have

q (x) := sup{|q (x)(v)| : q(v) ¤ 1} = 1.

Convention. Let q = 0 be a seminorm and let q(x) = 0. Then there exists a
v ∈ E with q(v) = 0, and we have q(x + tv) = |t| q(v), hence q (x)(±v) = q(v). So q
is not Gˆteaux di¬erentiable at x. Therefore, we call a seminorm smooth for some
a
di¬erentiability class, if and only if it is smooth on its carrier {x : q(x) > 0}.


13.4
130 Chapter III. Partitions of unity 13.5

13.5. Di¬erentiability properties of convex functions f can be translated in geo-
metric properties of Af :

Lemma. Di¬erentiability of convex functions. Let f : E ’ R be a contin-
uous convex function on a Banach space E, and let x0 ∈ E. Then the following
statements are equivalent:
(1) The function f is Gˆteaux (Fr´chet) di¬erentiable at x0 ;
a e
(2) There exists a unique ∈ E with

(v) ¤ f (x0 + v) ’ f (x0 ) for all v ∈ E;

(3) There exists a unique a¬ne hyperplane through (x0 , f (x0 )) which is tangent
to Af .
(4) The Minkowski functional of Af is Gˆteaux di¬erentiable at (x0 , f (x0 )).
a
Moreover, for a sublinear function f the following statements are equivalent:
(5) The function f is Gˆteaux (Fr´chet) di¬erentiable at x;
a e
(6) The point x0 (strongly) exposes the closed unit ball {x : f (x) ¤ 1}.
In particular, the following statements are equivalent for a convex function f :
(7) The function f is Gˆteaux (Fr´chet) di¬erentiable at x0 ;
a e
(8) The Minkowski functional of Af is Gˆteaux (Fr´chet) di¬erentiable at the
a e
point (x0 , f (x0 ));
(9) The point (x0 , f (x0 )) (strongly) exposes the polar (Af )o .

An element x— ∈ E — is said to expose a subset K ⊆ E if there exists a unique point
k0 ∈ K with x— (k0 ) = sup{x— (k) : k ∈ K}. It is said to strongly expose K, if
satis¬es in addition that x— (xn ) ’ x— (k0 ) implies xn ’ k0 .
By an a¬ne hyperplane H tangent to a convex set K at a point x ∈ K we mean
that x ∈ H and K lies on one side of H.

Proof. Let f be a convex function. By (13.4) and continuity we know that f is
Gˆteaux-di¬erentiable if and only if the sub-linear mapping f (x0 ) is linear. This
a
is exactly the case if f (x0 ) is minimal among all sub-linear mappings. From this
follows (1) ’ (2) by the following arguments: We have f (x0 )(v) ¤ f (x0 +v)’f (x0 ),
and (v) ¤ f (x0 + v) ’ f (x0 ) implies (v) ¤ f (x0 +t v)’f (x0 ) , and hence (v) ¤
t
f (x)(v).
(2) ’ (1) The uniqueness of implies f (x0 ) = since otherwise we had a linear
functional µ = 0 with u ¤ f (x) ’ . Then µ + contradicts uniqueness.
(2) ” (3) Any hyperplane tangent to Af at (x0 , f (x0 )) is described by a functional
( , s) ∈ E — R such that (x) + s t ≥ (x0 ) + s f (x0 ) for all t ≥ f (x). Note that
the scalar s cannot be 0, since this would imply that (x) ≥ (x0 ) for all x. It has
to be positive, since otherwise the left side would go to ’∞ for f (x) ¤ t ’ +∞.
Without loss of generality we may thus assume that s = 1, so the linear functional
is uniquely determined by the hyperplane. Moreover, (x ’ x0 ) ≥ f (x0 ) ’ f (x) or,
by replacing by ’ , we have (x0 + v) ≥ f (x0 ) + (v) for all v ∈ E.

13.5
13.6 13. Di¬erentiability of seminorms 131

(3) ” (4) Since the graph of a sublinear functional p is just the cone of {(y, 1) :
p(y) = 1}, the set Ap has exactly one tangent hyperplane at (x, 1) if and only if
the set {y : p(y) ¤ 1} has exactly one tangent hyperplane at x. Applying this to
the Minkowski-functional p of Af gives the desired result.
(5) ” (6) We show this ¬rst for Gˆteaux-di¬erentiability. We have to show that
a
there is a unique tangent hyperplane to x0 ∈ K := {x : f (x) ¤ 1} if and only if
x0 exposes K o := { ∈ E — : (x) ¤ 1 for all x ∈ K}. Let us assume 0 ∈ K and
0 = x0 ∈ ‚K. Then a tangent hyperplane to K at x0 is uniquely determined by
a linear functional ∈ E — with (x0 ) = 1 and (x) ¤ 1 for all x ∈ K. This is
equivalent to ∈ K o and (x0 ) = 1, since by Hahn-Banach there exists an ∈ K o
with (x0 ) = 1. From this the result follows.
This shows also (7) ” (8) ” (9) for Gˆteaux-di¬erentiability.
a
In order to show the statements for Fr´chet-di¬erentiability one has to show that
e
= f (x) is a Fr´chet derivative if and only if x0 is a strongly exposing point. This
e
is left to the reader, see also (13.19) for a more general result.

13.6. Lemma. Duality for convex functions. [Moreau, 1965].
: F — G ’ R be a dual pairing.
Let ,
(1) For f : F ’ R ∪ {+∞}, f = +∞ one de¬nes the dual function

f — : G ’ R ∪ {+∞}, f — (z) := sup{ z, y ’ f (y) : y ∈ F }.

(2) The dual function f — is convex and lower semi-continuous with respect to
the weak topology. Since a convex function g is lower semi-continuous if
and only if for all a ∈ R the set {x : g(x) > a} is open, equivalently the
convex set {x : g(x) ¤ a} is closed, this is equivalent for every topology
which is compatible with the duality.
— —
(3) f1 ¤ f2 ’ f1 ≥ f2 .
(4) f — ¤ g ” g — ¤ f .
(5) f —— = f if and only if f is lower semi-continuous and convex.
(6) Suppose z ∈ G satis¬es f (x + v) ≥ f (x) + z, v for all v (in particular, this
is true if z = f (x)). Then f (x) + f — (z) = z, x .
(7) If f1 (y) = f (y ’ a), then f1 (z) = z, a + f — (z).


(8) If f1 (y) = f (y) + a, then f1 (z) = f — (z) ’ a.


(9) If f1 (y) = f (y) + b, x , then f1 (z) = f — (z ’ b).


(10) If E = F = R and f ≥ 0 with f (0) = 0, then f — (t) = sup{ts ’ f (s) : t ≥ 0}
for t ≥ 0.
(11) If γ ≥ 0 is convex and γ(t) ’ 0, then γ(t) > 0 for t > 0.
t
(12) Let (F, G) be a Banach space and its dual. If γ ≥ 0 is convex and γ(0) = 0,
and f (y) := γ( y ), then f — (z) = γ — ( z ).
(13) A convex function f on a Banach space is Fr´chet di¬erentiable at a with
e
derivative b := f (a) if and only if there exists a convex non-negative func-
tion γ, with γ(0) = 0 and limt’0 γ(t) = 0, such that
t

f (a + h) ¤ f (a) + f (a), h + γ( h ).


13.6
132 Chapter III. Partitions of unity 13.6

Proof. (1) Since f = +∞, there is some y for which z, y ’ f (y) is ¬nite, hence
f — (z) > ’∞.
(2) The function z ’ z, y ’f (y) is continuous and linear, and hence the supremum
f — (z) is lower semi-continuous and convex. It remains to show that f — is not
constant +∞: This is not true. In fact, take f (t) = ’t2 then f — (s) = sup{s t’f (t) :
t ∈ R} = sup{s t + t2 : t ∈ R} = +∞. More generally, f — = +∞ ” f lies above
some a¬ne hyperplane, see (5).
— —
(3) If f1 ¤ f2 then z, y ’ f1 (z) ≥ z, y ’ f2 (z), and hence f1 (z) ≥ f2 (z).
(4) One has

∀z : f — (z) ¤ g(z) ” ∀z, y : z, y ’ f (y) ¤ g(z)
” ∀z, y : z, y ’ g(z) ¤ f (y)
” ∀y : g — (y) ¤ f (y).

(5) Since (f — )— is convex and lower semi-continuous, this is true for f provided
f = (f — )— . Conversely, let g(b) = ’a and g(z) = +∞ otherwise. Then g — (y) =
¤ f ” f — (b) ¤ ’a. If f is
sup{ z, y ’ g(z) : z ∈ G} = b, y + a. Hence, a + b,
convex and lower semi-continuous, then it is the supremum of all continuous linear
below it, and this is exactly the case if f — (b) ¤ ’a. Hence,
functionals a + b,
f —— (y) = sup{ z, y ’ f — (z) : z ∈ G} ≥ b, y + a and thus f = f —— .
(6) Let f (a+y) ≥ f (a)+ b, y . Then f — (b) = sup{ b, y ’f (y) : y ∈ F } = sup{ b, a+
v ’ f (a + v) : v ∈ F } ¤ sup{ b, a + b, v ’ f (a) ’ b, v : v ∈ F } = b, a ’ f (a).
(7) Let f1 (y) = f (y ’ a). Then

f1 (z) = sup{ z, y ’ f (y ’ a) : y ∈ F }
= sup{ z, y + a ’ f (y) : y ∈ F } = z, a + f — (z).

(8) Let f1 (y) = f (y) + a. Then

f1 (z) = sup{ z, y ’ f (y) ’ a : y ∈ F } = f — (z) ’ a.




(9) Let f1 (y) = f (y) + b, y . Then

f1 (z) = sup{ z, y ’ f (y) ’ b, y : y ∈ F }
= sup{ z ’ b, y ’ f (y) : y ∈ F } = f — (z ’ b).

(10) Let E = F = R and f ≥ 0 with f (0) = 0, and let s ≥ 0. Using that
s t ’ f (t) ¤ 0 for t ¤ 0 and that s 0 ’ f (0) = 0 we obtain

f — (s) = sup{s t ’ f (t) : t ∈ R} = sup{s t ’ f (t) : t ≥ 0}.

γ(t) γ(t)
(11) Let γ ≥ 0 with limt = 0, and let s > 0. Then there are t with s > t,
0 t
and hence
γ(t)
γ — (s) = sup{st ’ γ(t) : t ≥ 0} = sup{t(s ’ ) : t ≥ 0} > 0.
t
13.6
13.7 13. Di¬erentiability of seminorms 133

(12) Let f (y) = γ( y ). Then

f — (z) = sup{ z, y ’ γ( y ) : y ∈ F }
= sup{t z, y ’ γ(t) : y = 1, t ≥ 0}
= sup{sup{t z, y ’ γ(t) : y = 1}, t ≥ 0}
= sup{t z ’ γ(t) : y = 1, t ≥ 0}
= γ — ( z ).

(13) If f (a + h) ¤ f (a) + b, h + γ( h ), then we have

f (a + t h) ’ f (a) γ(t h )
¤ b, h + ,
t t
hence f (a)(h) ¤ b, h . Since h ’ f (a)(h) is sub-linear and the linear functionals
are minimal among the sublinear ones, we have equality. By convexity we have

f (a + t h) ’ f (a)
≥ b, h = f (a)(h).
t
So f is Fr´chet-di¬erentiable at a with derivative f (a)(h) = b, h , since the re-
e
mainder is bounded by γ( h ) which satis¬es γ( h ) ’ 0 for h ’ 0.
h


Conversely, assume that f is Fr´chet-di¬erentiable at a with derivative b. Then
e

|f (a + h) ’ f (a) ’ b, h |
’ 0 for h ’ 0,
h

and by convexity
g(h) := f (a + h) ’ f (a) ’ b, h ≥ 0.
Let γ(t) := sup{g(u) : u = |t|}. Since g is convex γ is convex, and obviously
γ(t) ∈ [0, +∞], γ(0) = 0 and γ(t) ’ 0 for t ’ 0. This is the required function.
t

13.7. Proposition. Continuity of the Fr´chet derivative. [Asplund, 1968].
e
The di¬erential f of any continuous convex function f on a Banach space is con-
tinuous on the set of all points where f is Fr´chet di¬erentiable. In general, it is
e
however neither uniformly continuous nor bounded, see (15.8).

Proof. Let f (x)(h) denote the one sided derivative. From convexity we conclude
≥ f (x) + f (x)(v). Suppose xn ’ x are points where f is Fr´chet
that f (x + v) e
Then we obtain f (xn )(v) ¤ f (xn + v) ’ f (xn ) which is bounded in
di¬erentiable.
n. Hence, the f (xn ) form a bounded sequence. We get

f (x) ≥ f (xn ), x ’ f — (f (xn )) since f (y) + f — (z) ≥ z, y
since f — (f (z)) + f (z) = f (z)(z)
= f (xn ), x + f (xn ) ’ f (xn ), xn
≥ f (xn ), x ’ xn + f (x) + f (x), xn ’ x since f (x + h) ≥ f (x) + f (x)(h)
= f (xn ) ’ f (x), x ’ xn + f (x).

13.7
134 Chapter III. Partitions of unity 13.8

Since xn ’ x and f (xn ) is bounded, both sides converge to f (x), hence

lim f (xn ), x ’ f — (f (xn )) = f (x).
n’∞


Since f is convex and Fr´chet-di¬erentiable at a := x with derivative b := f (x),
e
there exists by (13.6.13) a γ with

f (h) ¤ f (a) + b, h ’ a + γ( h ’ a ).

By duality we obtain using (13.6.3)

f — (z) ≥ z, a ’ f (a) + γ — ( z ’ b ).

If we apply this to z := f (xn ) we obtain

f — (f (xn )) ≥ f (xn ), x ’ f (x) + γ — ( f (xn ) ’ f (x) ).

Hence
γ — ( f (xn ) ’ f (x) ) ¤ f — (f (xn )) ’ f (xn ), x + f (x),

and since the right side converges to 0, we have that γ — ( f (xn ) ’ f (x) ) ’ 0.
Then f (xn ) ’ f (x) ’ 0 where we use that γ is convex, γ(0) = 0, and γ(t) > 0
for t > 0, thus γ is strictly monotone increasing.

13.8. Asplund spaces and generic Fr´chet di¬erentiability. From (13.4)
e
it follows easily that a convex function f : R ’ R is di¬erentiable at all except
countably many points. This has been generalized by [Rademacher, 1919] to: Ev-
ery Lipschitz mapping from an open subset of Rn to R is di¬erentiable almost
everywhere. Recall that a locally bounded convex function is locally Lipschitz, see
(13.2).

Proposition. For a Banach space E the following statements are equivalent:
(1) Every continuous convex function f : E ’ R is Fr´chet-di¬erentiable on a
e
dense Gδ -subset of E;
(2) Every continuous convex function f : E ’ R is Fr´chet-di¬erentiable on a
e
dense subset of E;
(3) Every locally Lipschitz function f : E ’ R is Fr´chet-di¬erentiable on a
e
dense subset of E;
(4) Every equivalent norm is Fr´chet-di¬erentiable at least at one point;
e
(5) E has no equivalent rough norm;
(6) Every (closed) separable subspace has a separable dual;
(7) The dual E — has the Radon-Nikodym property;
(8) Every linear mapping E ’ L1 (X, „¦, µ) which is integral is nuclear;
(9) Every closed convex bounded subset of E — is the closed convex hull of its
extremal points;
(10) Every bounded subset of E — is dentable.

13.8
13.8 13. Di¬erentiability of seminorms 135

A Banach space satisfying these equivalent conditions is called Asplund space.
Every Banach space with a Fr´chet di¬erentiable bump function is Asplund, [Eke-
e
land, Lebourg, 1976, p. 203]. It is an open question whether the converse is true.
Every WCG-space (i.e. a Banach space for which a weakly compact subset K exists,
whose linear hull is the whole space) is Asplund, [John, Zizler, 1976].
The Asplund property is inherited by subspaces, quotients, and short exact se-
quences, [Stegall, 1981].

About the proof. (1) [Asplund, 1968]: If a convex function is Fr´chet di¬eren-
e
tiable on a dense subset then it is so on a dense Gδ -subset, i.e. a dense countable
intersection of open subsets.
(2) is in fact a local property, since in [Borwein, Fitzpatrick, Kenderov, 1991] it
is mentioned that for a Lipschitz function f : E ’ R with Lipschitz constant L
de¬ned on a convex open set U the function

˜
f (x) := inf{f (y) + L x ’ y : y ∈ U }

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