Order summable

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements

(

x

i

)

i

=

1

∞

{\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}

in a preordered vector space

X

{\displaystyle X}

(that is,

x

i

≥

0

{\displaystyle x_{i}\geq 0}

for all

i

{\displaystyle i}

) is called order summable if

sup

n

=

1

,

2

,

…

∑

i

=

1

n

x

i

{\displaystyle \sup _{n=1,2,\ldots }\sum _{i=1}^{n}x_{i}}

exists in

X

{\displaystyle X}

.

For any

1

≤

p

≤

∞

{\displaystyle 1\leq p\leq \infty }

, we say that a sequence

(

x

i

)

i

=

1

∞

{\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}

of positive elements of

X

{\displaystyle X}

is of type

ℓ

p

{\displaystyle \ell ^{p}}

if there exists some

z

∈

X

{\displaystyle z\in X}

and some sequence

(

c

i

)

i

=

1

∞

{\displaystyle \left(c_{i}\right)_{i=1}^{\infty }}

in

ℓ

p

{\displaystyle \ell ^{p}}

such that

0

≤

x

i

≤

c

i

z

{\displaystyle 0\leq x_{i}\leq c_{i}z}

for all

i

{\displaystyle i}

.

The notion of order summable sequences is related to the completeness of the order topology.

See also

Ordered topological vector space

Order topology (functional analysis) – Topology of an ordered vector space

Ordered vector space – Vector space with a partial order

Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

References

Bibliography

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.

Schaefer, Helmut H.; Wolff, Manfred P. (1999).