<s>
In	O
mathematics	O
,	O
there	O
are	O
two	O
different	O
notions	O
of	O
semi-inner-product	B-Algorithm
.	O
</s>
<s>
This	O
article	O
will	O
deal	O
with	O
the	O
second	O
,	O
called	O
a	O
L-semi-inner	B-Algorithm
product	I-Algorithm
or	O
semi-inner	B-Algorithm
product	I-Algorithm
in	I-Algorithm
the	I-Algorithm
sense	I-Algorithm
of	I-Algorithm
Lumer	I-Algorithm
,	O
which	O
is	O
an	O
inner	O
product	O
not	O
required	O
to	O
be	O
conjugate	O
symmetric	O
.	O
</s>
<s>
It	O
was	O
formulated	O
by	O
Günter	O
Lumer	O
,	O
for	O
the	O
purpose	O
of	O
extending	O
Hilbert	O
space	O
type	O
arguments	O
to	O
Banach	O
spaces	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
We	O
mention	O
again	O
that	O
the	O
definition	O
presented	O
here	O
is	O
different	O
from	O
that	O
of	O
the	O
"	O
semi-inner	B-Algorithm
product	I-Algorithm
"	O
in	O
standard	O
functional	B-Application
analysis	I-Application
textbooks	O
,	O
where	O
a	O
"	O
semi-inner	B-Algorithm
product	I-Algorithm
"	O
satisfies	O
all	O
the	O
properties	O
of	O
inner	O
products	O
(	O
including	O
conjugate	O
symmetry	O
)	O
except	O
that	O
it	O
is	O
not	O
required	O
to	O
be	O
strictly	O
positive	O
.	O
</s>
<s>
Linearity	B-Architecture
in	O
the	O
1st	O
argument	O
,	O
meaning	O
:	O
</s>
<s>
Homogeneity	B-General_Concept
in	O
the	O
1st	O
argument	O
:	O
</s>
<s>
Conjugate	O
homogeneity	B-General_Concept
in	O
the	O
2nd	O
argument	O
:	O
</s>
<s>
A	O
semi-inner-product	B-Algorithm
is	O
different	O
from	O
inner	O
products	O
in	O
that	O
it	O
is	O
in	O
general	O
not	O
conjugate	O
symmetric	O
,	O
that	O
is	O
,	O
</s>
<s>
In	O
other	O
words	O
,	O
semi-inner-products	B-Algorithm
are	O
generally	O
nonlinear	O
about	O
its	O
second	O
variable	O
.	O
</s>
<s>
If	O
is	O
a	O
semi-inner-product	B-Algorithm
for	O
a	O
linear	O
vector	O
space	O
then	O
defines	O
a	O
norm	O
on	O
.	O
</s>
<s>
has	O
the	O
consistent	O
semi-inner-product	B-Algorithm
:	O
</s>
<s>
possesses	O
the	O
consistent	O
semi-inner-product	B-Algorithm
:	O
</s>
<s>
Following	O
the	O
idea	O
of	O
Lumer	O
,	O
semi-inner-products	B-Algorithm
were	O
widely	O
applied	O
to	O
study	O
bounded	O
linear	B-Architecture
operators	I-Architecture
on	O
Banach	O
spaces	O
.	O
</s>
<s>
In	O
2007	O
,	O
Der	O
and	O
Lee	O
applied	O
semi-inner-products	B-Algorithm
to	O
develop	O
large	O
margin	O
classification	O
in	O
Banach	O
spaces	O
.	O
</s>
<s>
Recently	O
,	O
semi-inner-products	B-Algorithm
have	O
been	O
used	O
as	O
the	O
main	O
tool	O
in	O
establishing	O
the	O
concept	O
of	O
reproducing	O
kernel	O
Banach	O
spaces	O
for	O
machine	O
learning	O
.	O
</s>
<s>
Semi-inner-products	B-Algorithm
can	O
also	O
be	O
used	O
to	O
establish	O
the	O
theory	O
of	O
frames	O
,	O
Riesz	O
bases	O
for	O
Banach	O
spaces	O
.	O
</s>
