Convex series

In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form

∑

i

=

1

∞

r

i

x

i

{\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}

where

x

1

,

x

2

,

…

{\displaystyle x_{1},x_{2},\ldots }

are all elements of a topological vector space

X

{\displaystyle X}

, and all

r

1

,

r

2

,

…

{\displaystyle r_{1},r_{2},\ldots }

are non-negative real numbers that sum to

1

{\displaystyle 1}

(that is, such that

∑

i

=

1

∞

r

i

=

1

{\displaystyle \sum _{i=1}^{\infty }r_{i}=1}

).

Types of Convex series

Suppose that

S

{\displaystyle S}

is a subset of

X

{\displaystyle X}

and

∑

i

=

1

∞

r

i

x

i

{\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}

is a convex series in

X

.

{\displaystyle X.}

If all

x

1

,

x

2

,

…

{\displaystyle x_{1},x_{2},\ldots }

belong to

S

{\displaystyle S}

then the convex series

∑

i

=

1

∞

r

i

x

i

{\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}

is called a convex series with elements of

S

{\displaystyle S}

.

If the set

{

x

1

,

x

2

,

…

}

{\displaystyle \left\{x_{1},x_{2},\ldots \right\}}

is a (von Neumann) bounded set then the series called a b-convex series.

The convex series

∑

i

=

1

∞

r

i

x

i

{\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}

is said to be a convergent series if the sequence of partial sums

(

∑

i

=

1

n

r

i

x

i

)

n

=

1

∞

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

converges in

X

{\displaystyle X}

to some element of

X

,

{\displaystyle X,}

which is called the sum of the convex series.

The convex series is called Cauchy if

∑

i

=

1

∞

r

i

x

i

{\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}

is a Cauchy series, which by definition means that the sequence of partial sums

(

∑

i

=

1

n

r

i

x

i

)

n

=

1

∞

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

is a Cauchy sequence.

Types of subsets

Convex series allow for the definition of special types of subsets that are well-behaved and useful with very good stability properties.

If

S

{\displaystyle S}

is a subset of a topological vector space

X

{\displaystyle X}

then

S

{\displaystyle S}

is said to be a:

cs-closed set if any convergent convex series with elements of

S

{\displaystyle S}

has its (each) sum in

S

.

{\displaystyle S.}

In this definition,

X

{\displaystyle X}

is not required to be Hausdorff, in which case the sum may not be unique. In any such case we require that every sum belong to

S

.