<s>
In	O
mathematics	O
specifically	O
,	O
in	O
functional	B-Application
analysis	I-Application
an	O
Asplund	B-Algorithm
space	I-Algorithm
or	O
strong	O
differentiability	O
space	O
is	O
a	O
type	O
of	O
well-behaved	O
Banach	O
space	O
.	O
</s>
<s>
Asplund	B-Algorithm
spaces	I-Algorithm
were	O
introduced	O
in	O
1968	O
by	O
the	O
mathematician	O
Edgar	O
Asplund	O
,	O
who	O
was	O
interested	O
in	O
the	O
Fréchet	O
differentiability	O
properties	O
of	O
Lipschitz	O
functions	O
on	O
Banach	O
spaces	O
.	O
</s>
<s>
There	O
are	O
many	O
equivalent	O
definitions	O
of	O
what	O
it	O
means	O
for	O
a	O
Banach	O
space	O
X	O
to	O
be	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
:	O
</s>
<s>
X	O
is	O
Asplund	O
if	O
,	O
and	O
only	O
if	O
,	O
every	O
non-empty	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
its	O
dual	O
space	O
X∗	O
has	O
weak-	O
∗	O
-slices	O
of	O
arbitrarily	O
small	O
diameter	O
.	O
</s>
<s>
In	O
1975	O
,	O
Huff	O
&	O
Morris	O
showed	O
that	O
this	O
property	O
is	O
equivalent	O
to	O
the	O
statement	O
that	O
every	O
bounded	B-Algorithm
,	O
closed	O
and	O
convex	O
subset	O
of	O
the	O
dual	O
space	O
X∗	O
is	O
closed	O
convex	O
hull	O
of	O
its	O
extreme	O
points	O
.	O
</s>
<s>
The	O
class	O
of	O
Asplund	B-Algorithm
spaces	I-Algorithm
is	O
closed	O
under	O
topological	O
isomorphisms	O
:	O
that	O
is	O
,	O
if	O
X	O
and	O
Y	O
are	O
Banach	O
spaces	O
,	O
X	O
is	O
Asplund	O
,	O
and	O
X	O
is	O
homeomorphic	O
to	O
Y	O
,	O
then	O
Y	O
is	O
also	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
closed	O
linear	O
subspace	O
of	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
is	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
quotient	O
space	O
of	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
is	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
class	O
of	O
Asplund	B-Algorithm
spaces	I-Algorithm
is	O
closed	O
under	O
extensions	O
:	O
if	O
X	O
is	O
a	O
Banach	O
space	O
and	O
Y	O
is	O
an	O
Asplund	O
subspace	O
of	O
X	O
for	O
which	O
the	O
quotient	O
space	O
X⁄Y	O
is	O
Asplund	O
,	O
then	O
X	O
is	O
Asplund	O
.	O
</s>
<s>
Every	O
locally	O
Lipschitz	O
function	O
on	O
an	O
open	O
subset	O
of	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
is	O
Fréchet	O
differentiable	O
at	O
the	O
points	O
of	O
some	O
dense	O
subset	O
of	O
its	O
domain	O
.	O
</s>
<s>
The	O
following	O
theorem	O
from	O
Asplund	O
's	O
original	O
1968	O
paper	O
is	O
a	O
good	O
example	O
of	O
why	O
non-Asplund	O
spaces	O
are	O
badly	O
behaved	O
:	O
if	O
X	O
is	O
not	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
,	O
then	O
there	O
exists	O
an	O
equivalent	O
norm	O
on	O
X	O
that	O
fails	O
to	O
be	O
Fréchet	O
differentiable	O
at	O
every	O
point	O
of	O
X	O
.	O
</s>
<s>
In	O
1976	O
,	O
Ekeland	O
&	O
Lebourg	O
showed	O
that	O
if	O
X	O
is	O
a	O
Banach	O
space	O
that	O
has	O
an	O
equivalent	O
norm	O
that	O
is	O
Fréchet	O
differentiable	O
away	O
from	O
the	O
origin	O
,	O
then	O
X	O
is	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
However	O
,	O
in	O
1990	O
,	O
Haydon	O
gave	O
an	O
example	O
of	O
an	O
Asplund	B-Algorithm
space	I-Algorithm
that	O
does	O
not	O
have	O
an	O
equivalent	O
norm	O
that	O
is	O
Gateaux	B-Algorithm
differentiable	I-Algorithm
away	O
from	O
the	O
origin	O
.	O
</s>
