Definition
edit
Let
(
X
i
)
i
∈
I
{\displaystyle (X_{i})_{i\in I}}
be a family of Banach spaces, where
I
{\displaystyle I}
may have arbitrarily large cardinality. Set
P
:=
∏
i
∈
I
X
i
,
{\displaystyle P:=\prod _{i\in I}X_{i},}
the product vector space.
The index set
I
{\displaystyle I}
becomes a measure space when endowed with its counting measure (which we shall denote by
μ
{\displaystyle \mu }
), and each element
(
x
i
)
i
∈
I
∈
P
{\displaystyle (x_{i})_{i\in I}\in P}
induces a function
I
→
R
,
i
↦
‖
x
i
‖
.
{\displaystyle I\to \mathbb {R} ,i\mapsto \|x_{i}\|.}
Thus, we may define a function
Φ
:
P
→
R
∪
{
∞
}
,
(
x
i
)
i
∈
I
↦
∫
I
‖
x
i
‖
p
d
μ
(
i
)
{\displaystyle \Phi :P\to \mathbb {R} \cup \{\infty \},(x_{i})_{i\in I}\mapsto \int _{I}\|x_{i}\|^{p}\,d\mu (i)}
and we then set
⨁
⨁
p
i
∈
I
X
i
:=
{
(
x
i
)
i
∈
I
∈
P
∣
Φ
(
(
x
i
)
i
∈
I
)
<
∞
}
{\displaystyle \sideset {}{^{p}}\bigoplus \limits _{i\in I}X_{i}:=\{(x_{i})_{i\in I}\in P\mid \Phi ((x_{i})_{i\in I})<\infty \}}
together with the norm
‖
(
x
i
)
i
∈
I
‖
:=
(
∫
i
∈
I
‖
x
i
‖
p
d
μ
(
i
)
)
1
/
p
.
{\displaystyle \|(x_{i})_{i\in I}\|:=\left(\int _{i\in I}\|x_{i}\|^{p}\,d\mu (i)\right)^{1/p}.}
The result is a normed Banach space, and this is precisely the Lp sum of
(
X
i
)
i
∈
I
.
{\displaystyle (X_{i})_{i\in I}.}
Properties
edit
Whenever infinitely many of the
X
i
{\displaystyle X_{i}}
contain a nonzero element, the topology induced by the above norm is strictly in between product and box topology.
Whenever infinitely many of the
X
i
{\displaystyle X_{i}}
contain a nonzero element, the Lp sum is neither a product nor a coproduct .
References
edit
^ Helemskii, A. Ya. (2006). Lectures and Exercises on Functional Analysis . Translations of Mathematical Monographs. American Mathematical Society. ISBN 0-8218-4098-3 .