Fréchet space

Fréchet space

We consider two classes of

topological vector spaces

, one more general
than the other. Following Rudin

[

1

]

we will define a Fréchet space
to be an element of the smaller class, and refer to an

instance

of the
more general class as an

F-space

. After giving the

definitions

, we will explain why one definition is

stronger

than the
other.

Definition 1.

An F-space is a complete topological vector space whose

topology

is
induced by a

translation

invariant metric. To be more precise, we say
that

U

is an F-space if there exists a metric

function

d

:

U

×

U

→

ℝ

such that

d

⁢

(

x

,

y

)

=

d

⁢

(

x

+

z

,

y

+

z

)

,

x

,

y

,

z

∈

U

;

and such that the

collection

of balls

B

ϵ

⁢

(

x

)

=

{

y

∈

U

:

d

⁢

(

x

,

y

)

<

ϵ

}

,

x

∈

U

,

ϵ

>

0

is a base for the topology of

U

.

Note 1.

Recall that a topological vector space is a

uniform space

.
The

hypothesis

that

U

is

complete

is formulated in reference to this
uniform structure. To be more precise, we say that a

sequence

a

n

∈

U

,

n

=

1

,

2

,

…

is Cauchy if for every

neighborhood

O

of the
origin there exists an

N

∈

ℕ

such that

a

n

-

a

m

∈

O

for all

n

,

m

>

N

. The completeness condition then takes
the usual form of the hypothesis that all

Cauchy sequences

possess a

limit point

.

Note 2.

It is customary to include the hypothesis that

U

is

Hausdorff

in the definition of a topological vector space.
Consequently, a Cauchy sequence in a complete topological space
will have a unique limit.

Note 3.

Since

U

is assumed to be complete, the pair

(

U

,

d

)

is a

complete metric space

. Thus, an

equivalent

definition of
an F-space is that of a

vector space

equipped with a complete, translation-invariant (but not necessarily

homogeneous

(

http://planetmath.org/NormedVectorSpace

)) metric, such that the

operations

of

scalar
multiplication

and

vector addition

are

continuous

with respect to this
metric.

Definition 2.

A Fréchet space is a complete topological vector space (either real
or complex) whose topology is induced by a

countable

family of
semi-norms. To be more precise, there exist semi-norm functions

∥

-

∥

n

:

U

→

ℝ

,

n

∈

ℕ

,

such that the collection of all balls

B

ϵ

(

n

)

⁢

(

x

)

=

{

y

∈

U

:

∥

x

-

y

∥

n

<

ϵ

}

,

x

∈

U

,

ϵ

>

0

,

n

∈

ℕ

,

is a base for the topology of

U

.

Proposition 1

Let

U

be a complete topological vector space. Then,

U

is a
Fréchet space if and only if it is a locally convex F-space.

Proof.

First, let us show that a Fréchet space is a locally convex F-space,
and then prove the

converse

. Suppose then that

U

is Fréchet. The
semi-norm balls are convex; this follows directly from the semi-norm
axioms. Therefore

U

is locally convex. To obtain the desired
distance function we set

d

⁢

(

x

,

y

)

=

∑

n

=

0

∞

2

-

n

⁢

∥

x

-

y

∥

n

1

+

∥

x

-

y

∥

n

,

x

,

y

∈

U

.

(1)

We now show that

d

satisfies

the metric axioms. Let

x

,

y

∈

U

such that

x

≠

y

be given. Since

U

is Hausdorff, there is at
least one

seminorm

such

∥

x

-

y

∥

n

>

0

.

Hence

d

⁢

(

x

,

y

)

>

0

.

Let

a

,

b

,

c

>

0

be three

real numbers

such that

a

≤

b

+

c

.

A straightforward calculation shows
that

a

1

+

a

≤

b

1

+

b

+

c

1

+

c

,

(2)

as well. The above trick underlies the definition (

1

) of
our metric function. By the seminorm axioms we have that

∥

x

-

z

∥

n

≤

∥

x

-

y

∥

n

+

∥

y

-

z

∥

n

,

x

,

y

,

z

∈

U

for all

n

. Combining this with (

1

) and
(

2

) yields the

triangle inequality

for

d

.

Next let us suppose that

U

is a locally convex F-space, and prove
that it is Fréchet. For every

n

=

1

,

2

,

…

let

U

n

be an open
convex neighborhood of the origin,

contained

inside a ball of radius

1

/

n

about the origin. Let

∥

-

∥

n

be the seminorm with

U

n

as the

unit ball

. By definition, the unit balls of these
seminorms give a

neighborhood base

for the topology of

U

. QED.

References

1

W.Rudin,

Functional Analysis

.

Title

Fréchet space

Canonical

name

FrechetSpace

Date of creation

2013-03-22 13:06:10

Last modified on

2013-03-22 13:06:10

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

51

Author

rmilson (146)

Entry type

Definition

Classification

msc 57N17

Classification

msc 54E50

Classification

msc 52A07

Related topic

TopologicalVectorSpace

Related topic

HausdorffSpaceNotCompletelyHausdorff

Defines

F-space

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