aarjaneiro/planetmath huggingface dataset:
normal family

normal family

Definition

.

A set (sometimes called a family)

ℱ

of

continuous functions

f

:

X

→

Y

for some (

complete

(

http://planetmath.org/Complete

))

metric spaces

X

and

Y

is called if each sequence of

functions

in

ℱ

contains a

subsequence

which converges uniformly on

compact subsets

of

X

to a
continuous function from

X

to

Y

.

This

definition

is often used in

complex analysis

for

spaces of holomorphic functions

. It turns out that a sequence of

holomorphic functions

that converges uniformly on compact sets

converges

to a holomorphic function. So you can replace

X

with a region in the

complex plane

,

Y

with the complex plane itself
and every instance of “

continuous

” with “holomorphic” and you get a version of the definition most used in complex analysis.

Another space where this is often used is the space of meromorphic functions.
This is similar to the holomorphic case, but instead of using the

standard
metric

for convergence we must use the

spherical metric

. That is if

σ

is the spherical metric, then want

f

n

⁢

(

z

)

→

f

⁢

(

z

)

uniformly on compact
subsets to mean that

σ

⁢

(

f

n

⁢

(

z

)

,

f

⁢

(

z

)

)

goes to 0 uniformly on compact
subsets.

Note that this is a classical definition that, while very often used, is not really
with modern naming. In more modern ,
one would give a metric
on the space of continuous (holomorphic) functions that corresponds to
convergence on compact subsets and then you’d say “

precompact set

of functions” in such a metric space instead of saying “ family of continuous (holomorphic) functions”. This added generality however
makes it more cumbersome to use since one would need to define the metric mentioned above.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

normal family

Canonical

name

NormalFamily

Date of creation

2013-03-22 14:17:49

Last modified on

2013-03-22 14:17:49

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 30A99

Generated on Fri Feb 9 20:13:56 2018 by

LaTeXML
open mapping theorem

open mapping theorem

There are two important theorems having this .

∙

In the

context

of

functions

of a

complex variable

:

Theorem.

Every non-constant

analytic function

on a region is an open mapping.

∙

In the context of

functional analysis

:

Theorem.

Every surjective

continuous linear mapping

between two

Banach spaces

is an open mapping.

Title

open mapping theorem

Canonical name

OpenMappingTheorem

Date of creation

2013-03-22 13:12:41

Last modified on

2013-03-22 13:12:41

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

14

Author

Koro (127)

Entry type

Theorem

Classification

msc 30A99

Classification

msc 46A30

Related topic

ClosedGraphTheorem

Generated on Fri Feb 9 20:13:57 2018 by

LaTeXML
orientation

orientation

Let

α

be a

rectifiable

,

Jordan curve

in

ℝ

2

and

z

0

be a point in

ℝ

2

-

Im

⁡

(

α

)

and let

α

have a

winding number

W

[

α

:

z

0

]

. Then

W

[

α

:

z

0

]

=

±

1

; all points inside

α

will have the same index and we define the

orientation

of a Jordan curve

α

by saying that

α

is

positively oriented

if the index of every point in

α

is

+

1

and

negatively oriented

if it is

-

1

.

Title

orientation

Canonical name

Orientation

Date of creation

2013-03-22 12:56:09

Last modified on

2013-03-22 12:56:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 30A99

Related topic

SensePreservingMapping

Generated on Fri Feb 9 20:13:59 2018 by

LaTeXML
persistence of analytic relations

persistence of analytic relations

The principle of

persistence of analytic relations

states that any algebraic relation between several

analytic functions

which holds on a sufficiently large set also holds wherever the

functions

are defined.

A more explicit statement of this principle is as follows: Let

f

1

,

f

2

,

…

⁢

f

n

be complex analytic functions. Suppose that there exists an

open set

D

on which all these functions are defined and that there exists a

polynomial

p

of

n

variables such that

p

⁢

(

f

1

⁢

(

z

)

,

f

2

⁢

(

z

)

,

…

,

f

n

⁢

(

z

)

)

=

0

whenever

z

lies in a subset

X

of

D

which has a

limit point

in

D

. Then

p

⁢

(

f

1

⁢

(

z

)

,

f

2

⁢

(

z

)

,

…

⁢

f

n

⁢

(

z

)

)

=

0

for all

z

∈

D

.

This fact is a simple consequence of the

rigidity theorem for analytic functions

. If

f

1

,

f

2

,

…

⁢

f

n

are all analytic in

D

, then

p

⁢

(

f

1

⁢

(

z

)

,

f

2

⁢

(

z

)

,

…

⁢

f

n

⁢

(

z

)

)

is also analytic in

D

. Hence, if

p

⁢

(

f

1

⁢

(

z

)

,

f

2

⁢

(

z

)

,

…

,

f

n

⁢

(

z

)

)

=

0

when

z

in

X

, then

p

⁢

(

f

1

⁢

(

z

)

,

f

2

⁢

(

z

)

,

…

,

f

n

⁢

(

z

)

)

=

0

for all

z

∈

D

.

This principle is very useful in establishing identites involving analytic functions because it means that it suffices to show that the identity holds on a small subset. For instance, from the fact that the familiar identity

sin

2

⁡

x

+

cos

2

⁡

x

=

1

holds for all real

x

, it automatically holds for all

complex

values of

x

. This principle also means that it is unnecessary to specify for which values of the variable an algebraic relation between analytic functions holds since, if such a relation holds, it will hold for all values for which the functions appearing in the relation are defined.

Title

persistence of analytic relations

Canonical name

PersistenceOfAnalyticRelations

Date of creation

2013-03-22 14:44:17

Last modified on

2013-03-22 14:44:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30A99

Related topic

ComplexSineAndCosine

Generated on Fri Feb 9 20:14:01 2018 by

LaTeXML
persistence of differential equations

persistence of differential equations

The

persistence of analytic relations

has important consequences
for the theory of

differential equations

in the

complex plane

.
Suppose that a

function

f

satisfies a differential equation

F

⁢

(

x

,

f

⁢

(

x

)

,

f

′

⁢

(

x

)

,

…

,

f

(

n

)

)

=

0

where

F

is a

polynomial

.
This equation may be viewed as a polynomial relation between the

n

+

2

functions

id

,

f

,

f

′

,

…

,

f

(

n

)

hence, by the
persistence of analytic relations, it will also hold for the

analytic
continuations

of these functions. In other words, if an
algebraic differential equation holds for a function in some
region, it will still hold when that function is analytically
continued to a larger region.

An interesting special case is that of the

homogeneous linear
differential equation

with polynomial coefficients. In that
case, we have the principle of superposition which guarantees
that a

linear combination

of solutions is also a solution.
Hence, if we start with a basis of solutions to our equation
about some point and analytically continue them back to our
starting point, we obtain linear combinations of those
solutions. This observation plays a very important role in the
theory of differential equations in the complex plane and is
the foundation for the notion of

monodromy group

and Riemann’s
global characterization of the

hypergeometric function

.

For a less exalted illustrative example, we can consider the

complex logarithm

. The differential equation

x

⁢

y

′′

+

y

′

=

0

has as solutions

y

=

1

and

y

=

log

⁡

x

. While the former
is as singly valued as functions get, the latter is multiply
valued. Hence upon performong analytic continuation, we expect
that the second solution will continue to a linear combination
of the two solutions. This, of course is exactly what happens;
upon analytic continuation, the second solution becomes the
solution

y

=

log

⁡

x

+

n

⁢

π

⁢

i

where

n

is an integer whose
value depends on how we carry out the analytic continuation.

Title

persistence of differential equations

Canonical name

PersistenceOfDifferentialEquations

Date of creation

2013-03-22 16:20:35

Last modified on

2013-03-22 16:20:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Corollary

Classification

msc 30A99

Generated on Fri Feb 9 20:14:03 2018 by

LaTeXML
places of holomorphic function

places of holomorphic function

If

c

is a

complex

constant

and

f

a

holomorphic function

in a domain

D

of

ℂ

, then

f

has in every

compact

(closed (

http://planetmath.org/TopologyOfTheComplexPlane

) and

bounded

(

http://planetmath.org/Bounded

)) subdomain of

D

at most a

finite set

of

http://planetmath.org/node/9084

c

-places, i.e. the points

z

where

f

⁢

(

z

)

=

c

,  except when

f

⁢

(

z

)

≡

c

in the whole

D

.

Proof.

Let

A

be a subdomain of

D

.  Suppose that there is an

infinite

amount of

c

-places of

f

in

A

.  By

http://planetmath.org/node/2125

Bolzano–Weierstrass theorem, these

c

-places have an

accumulation point

z

0

, which belongs to the

closed set

A

.  Define the

constant function

g

such that

g

⁢

(

z

)

=

c

for all

z

in

D

.  Then

g

is holomorphic in the domain

D

and

g

⁢

(

z

)

=

c

in an infinite subset of

D

with the accumulation point

z

0

.  Thus in the

c

-places of

f

we have

g

⁢

(

z

)

=

f

⁢

(

z

)

.

Consequently, the

identity theorem of holomorphic functions

implies that

f

⁢

(

z

)

=

g

⁢

(

z

)

=

c

in the whole

D

.  Q.E.D.

Title

places of holomorphic function

Canonical name

PlacesOfHolomorphicFunction

Date of creation

2013-03-22 18:54:18

Last modified on

2013-03-22 18:54:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Corollary

Classification

msc 30A99

Related topic

ZerosAndPolesOfRationalFunction

Related topic

IdentityTheorem

Related topic

TopologyOfTheComplexPlane

Generated on Fri Feb 9 20:14:05 2018 by

LaTeXML
proof of fundamental theorem of algebra

proof of fundamental theorem of algebra

If

f

⁢

(

x

)

∈

ℂ

⁢

[

x

]

let

a

be a root of

f

⁢

(

x

)

in some

extension

of

ℂ

. Let

K

be a

Galois closure

of

ℂ

⁢

(

a

)

over

ℝ

and set

G

=

Gal

⁡

(

K

/

ℝ

)

.
Let

H

be a Sylow 2-subgroup of

G

and let

L

=

K

H

(the

fixed field

of

H

in

K

).
By the

Fundamental Theorem of Galois Theory

we have

[

L

:

ℝ

]

=

[

G

:

H

]

, an

odd number

. We may write

L

=

ℝ

⁢

(

b

)

for some

b

∈

L

, so the

minimal polynomial

m

b

,

ℝ

⁢

(

x

)

is

irreducible

over

ℝ

and of odd
degree. That degree must be 1, and hence

L

=

ℝ

, which
means that

G

=

H

, a 2-group. Thus

G

1

=

Gal

⁡

(

K

/

ℂ

)

is also a 2-group. If

G

1

≠

1

choose

G

2

≤

G

1

such that

[

G

1

:

G

2

]

=

2

, and set

M

=

K

G

2

,
so that

[

M

:

ℂ

]

=

[

G

1

:

G

2

]

=

2

. But any

polynomial

of
degree 2 over

ℂ

has roots in

ℂ

by the

quadratic formula

, so such a field

M

cannot exist. This

contradiction

shows that

G

1

=

1

. Hence

K

=

ℂ

and

a

∈

ℂ

, completing the proof.

Title

proof of fundamental theorem of algebra

Canonical name

ProofOfFundamentalTheoremOfAlgebra

Date of creation

2013-03-22 13:09:39

Last modified on

2013-03-22 13:09:39

Owner

scanez (1021)

Last modified by

scanez (1021)

Numerical id

5

Author

scanez (1021)

Entry type

Proof

Classification

msc 30A99

Classification

msc 12D99

Generated on Fri Feb 9 20:14:12 2018 by

LaTeXML
proof of fundamental theorem of algebra (Rouché’s theorem)

proof of fundamental theorem of algebra (Rouché’s theorem)

The

fundamental theorem of algebra

can be proven using Rouché’s theorem. Not only is this proof interesting because it demonstrates an important result, it also serves to provide an example of how to use Rouché’s theorem. Since it is quite , it can be thought of as a “toy model” (see toy theorem) for theorems on the zeroes of

analytic functions

. For a variant of this proof in of the

argument principle

(of which Rouché’s theorem is a consequence), please see the proof of the fundamental theorem of algebra (argument principle) (

http://planetmath.org/ProofOfFundamentalTheoremOfAlgebra3

).

Proof.

Let

n

denote the degree of

f

. Without loss of generality, the assumption can be made that the

leading coefficient

of

f

is

1

. Thus,

f

⁢

(

z

)

=

z

n

+

∑

m

=

0

n

-

1

c

m

⁢

z

m

.

Let

R

=

1

+

∑

m

=

0

n

-

1

|

c

m

|

. Note that, by choice of

R

, whenever

|

z

|

>

R

,

f

⁢

(

z

)

≠

0

. Suppose that

|

z

|

≥

R

. Since

R

≥

1

,

|

z

a

|

≤

|

z

b

|

whenever

0

<

a

<

b

. Hence, we have the following of

inequalities

:

|

∑

m

=

0

n

-

1

c

m

⁢

z

m

|

≤

1

+

∑

m

=

0

n

-

1

|

c

m

|

⁢

|

z

m

|

≤

|

z

n

-

1

|

+

∑

m

=

0

n

-

1

|

c

m

|

⁢

|

z

n

-

1

|

≤

R

⁢

|

z

n

-

1

|

≤

|

z

n

|

Since

polynomials

in

z

are entire, they are certainly analytic functions in the disk

|

z

|

≤

R

. Thus, Rouché’s theorem can be applied to them. Since

|

∑

m

=

0

n

-

1

c

m

⁢

z

m

|

≤

|

z

n

|

for

|

z

|

≥

R

, Rouché’s theorem yields that

z

n

and

f

⁢

(

z

)

have the same number of zeroes in the disk

|

z

|

≤

R

. Since

z

n

has a single zero of

multiplicity

n

at

z

=

0

, which counts as

n

zeroes,

f

⁢

(

z

)

must also have

n

zeroes counted according to multiplicity in the disk

|

z

|

≤

R

. By choice of

R

, it follows that

f

has exactly

n

zeroes in the

complex plane

.
∎

Title

proof of fundamental theorem of algebra

(Rouché’s theorem)

Canonical name

ProofOfFundamentalTheoremOfAlgebraRouchesTheorem

Date of creation

2013-03-22 14:36:21

Last modified on

2013-03-22 14:36:21

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 30A99

Classification

msc 12D99

Generated on Fri Feb 9 20:14:15 2018 by

LaTeXML
proof of fundamental theorem of algebra (argument principle)

proof of fundamental theorem of algebra (argument principle)

The

fundamental theorem of algebra

can be proven using the

argument principle

. Not only is this proof interesting because it demonstrates an important result, it also serves as an example of how to use the argument principle. Since it is so simple, it can be thought of as a “toy model” (see toy theorem) for theorems on the zeros of

analytic functions

. For a variant of this proof using Rouché’s theorem (which is a consequence of the argument principle) please see the proof of the fundamental theorem of algebra (Rouché’s theorem).

Proof.

Consider the

rational function

g

⁢

(

z

)

=

z

⁢

f

′

⁢

(

z

)

f

⁢

(

z

)

.

Denote the degree of the

polynomial

f

by

n

. Then we can write

g

⁢

(

z

)

=

n

⁢

z

n

+

lower degree terms

z

n

+

lower degree terms

.

This makes it clear that

lim

z

→

∞

⁡

g

⁢

(

z

)

=

n

. Hence there exists a real constant

R

such that

|

g

⁢

(

z

)

-

n

|

<

1

/

2

whenever

|

z

|

≥

R

.

Consider the

integral

I

=

∮

|

z

|

=

R

f

′

⁢

(

z

)

f

⁢

(

z

)

⁢

𝑑

z

.

This can be rewritten as

I

=

∮

|

z

|

=

R

g

⁢

(

z

)

z

⁢

𝑑

z

.

Split the integral into two parts, writing

I

=

I

1

+

I

2

where

I

1

=

∮

|

z

|

=

R

n

z

⁢

𝑑

z

,

I

2

=

∮

|

z

|

=

R

g

⁢

(

z

)

-

n

z

⁢

|

d

⁢

z

|

.

The integral

I

1

is easy:

I

1

=

2

⁢

π

⁢

i

⁢

n

. As for

I

2

, we shall bound it using our bound for

|

g

⁢

(

z

)

-

n

|

.

|

I

2

|

≤

∮

|

z

|

=

R

|

g

⁢

(

z

)

-

n

|

|

z

|

⁢

𝑑

z

≤

1

2

⁢

∮

|

d

⁢

z

|

|

z

|

=

π

.

Since polynomials are analytic functions in the whole

complex plane

,

f

is an analytic function of

z

when

|

z

|

≤

R

, so the argument principle applies and we conclude that

I

/

2

⁢

π

⁢

i

must equal the number of zeros of

f

, counted with

multiplicity

. Among other things, this means that

I

/

2

⁢

π

⁢

i

must be an

integer

. By explicit computation, we already know that

I

1

/

2

⁢

π

⁢

i

is also an integer. Hence

|

I

/

(

2

⁢

π

⁢

i

)

-

I

1

/

(

2

⁢

π

⁢

i

)

|

is an integer. But

|

I

(

2

⁢

π

⁢

i

)

-

I

1

(

2

⁢

π

⁢

i

)

|

=

|

I

2

(

2

⁢

π

⁢

i

)

|

≤

1

2

.

Now, the only integer smaller that

1

/

2

in

absolute value

is

0

, so we must have

I

=

I

1

. This implies that

f

has

n

zeros (counting with multiplicity) when

|

z

|

<

R

. (By the way we chose

R

,

f

⁢

(

z

)

≠

0

whenever

z

≥

R

, so

f

has exactly n zeros in the whole complex plane.)

Title

proof of fundamental theorem of algebra

(argument principle)

Canonical name

ProofOfFundamentalTheoremOfAlgebraargumentPrinciple

Date of creation

2013-03-22 14:36:14

Last modified on

2013-03-22 14:36:14

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30A99

Classification

msc 12D99

Generated on Fri Feb 9 20:14:18 2018 by

LaTeXML
proof of limit rule of product

proof of limit rule of product

Let

f

and

g

be real (

http://planetmath.org/RealFunction

) or

complex functions

having the limits

lim

x

→

x

0

⁡

f

⁢

(

x

)

=

F

and

lim

x

→

x

0

⁡

g

⁢

(

x

)

=

G

.

Then also the limit

lim

x

→

x

0

⁡

f

⁢

(

x

)

⁢

g

⁢

(

x

)

exists and equals

F

⁢

G

.

Proof.

Let

ε

be any positive number.  The

assumptions

imply the

existence

of the positive numbers

δ

1

,

δ

2

,

δ

3

such that

|

f

⁢

(

x

)

-

F

|

<

ε

2

⁢

(

1

+

|

G

|

)

⁢

when

⁢

0

<

|

x

-

x

0

|

<

δ

1

(1)

|

g

⁢

(

x

)

-

G

|

<

ε

2

⁢

(

1

+

|

F

|

)

⁢

when

⁢

0

<

|

x

-

x

0

|

<

δ

2

,

(2)

|

g

⁢

(

x

)

-

G

|

<

1

⁢

when

⁢

0

<

|

x

-

x

0

|

<

δ

3

.

(3)

According to the condition (3) we see that

|

g

⁢

(

x

)

|

=

|

g

⁢

(

x

)

-

G

+

G

|

≦

|

g

⁢

(

x

)

-

G

|

+

|

G

|

<

1

+

|

G

|

⁢

when

⁢

0

<

|

x

-

x

0

|

<

δ

3

.

Supposing then that

0

<

|

x

-

x

0

|

<

min

⁡

{

δ

1

,

δ

2

,

δ

3

}

and using (1) and (2) we obtain

|

f

⁢

(

x

)

⁢

g

⁢

(

x

)

-

F

⁢

G

|

=

|

f

⁢

(

x

)

⁢

g

⁢

(

x

)

-

F

⁢

g

⁢

(

x

)

+

F

⁢

g

⁢

(

x

)

-

F

⁢

G

|

≦

|

f

⁢

(

x

)

⁢

g

⁢

(

x

)

-

F

⁢

g

⁢

(

x

)

|

+

|

F

⁢

g

⁢

(

x

)

-

F

⁢

G

|

=

|

g

⁢

(

x

)

|

⋅

|

f

⁢

(

x

)

-

F

|

+

|

F

|

⋅

|

g

⁢

(

x

)

-

G

|

<

(

1

+

|

G

|

)

⁢

ε

2

⁢

(

1

+

|

G

|

)

+

(

1

+

|

F

|

)

⁢

ε

2

⁢

(

1

+

|

F

|

)

=

ε

This settles the proof.

Title

proof of limit rule of product

Canonical name

ProofOfLimitRuleOfProduct

Date of creation

2013-03-22 17:52:22

Last modified on

2013-03-22 17:52:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Proof

Classification

msc 30A99

Classification

msc 26A06

Related topic

ProductOfFunctions

Related topic

TriangleInequality

Related topic

ProductAndQuotientOfFunctionsSum

Generated on Fri Feb 9 20:14:34 2018 by

LaTeXML
proof of open mapping theorem

proof of open mapping theorem

We prove that if

Λ

:

X

→

Y

is a

continuous

linear surjective map between

Banach spaces

, then

Λ

is an open map. It suffices to show that

Λ

maps the open

unit ball

in

X

to a

neighborhood

of the origin of

Y

.

Let

U

,

V

be the open unit balls in

X

,

Y

respectively. Then

X

=

∪

k

∈

ℕ

k

⁢

U

, so, since

Λ

is surjective,

Y

=

Λ

⁢

(

X

)

=

Λ

⁢

(

∪

k

∈

ℕ

k

⁢

U

)

=

∪

k

∈

ℕ

Λ

⁢

(

k

⁢

U

)

. By the

Baire category theorem

,

Y

is not the union of countably many nowhere

dense sets

, so there is some

k

∈

ℕ

and some

open set

W

⊂

Y

such that

W

is contained in the

closure

of

Λ

⁢

(

k

⁢

U

)

.

Let

y

0

∈

W

, and pick

η

>

0

so that

y

0

+

y

∈

W

for all

y

with

||

y

||

<

η

. Then

y

0

and

y

0

+

y

are

limit points

of

Λ

⁢

(

k

⁢

U

)

, so there are sequences

x

i

′

and

x

i

′′

in

k

⁢

U

with

Λ

⁢

x

i

′

→

y

0

and

Λ

⁢

x

i

′′

→

y

0

+

y

. Letting

x

i

=

x

i

′′

-

x

i

′

, we have

||

x

i

||

<

2

⁢

k

and

Λ

⁢

x

i

→

y

. So for any

y

∈

η

⁢

V

there is a sequence

x

i

in

2

⁢

k

⁢

U

with

Λ

⁢

x

i

→

y

. Then by the linearity of

Λ

, we have that for any

ϵ

>

0

and any

y

∈

Y

, there is an

x

∈

X

with:

||

x

||

<

δ

-

1

⁢

||

y

||

and

||

Λ

⁢

x

-

y

||

<

ϵ

(1)

where

δ

=

η

/

2

⁢

k

.

Now let

y

∈

δ

⁢

V

and

ϵ

>

0

. Then there is some

x

1

with

||

x

1

||

<

1

and

||

y

-

Λ

⁢

x

1

||

<

ϵ

⁢

δ

. Define a sequence

x

n

inductively as follows. Assume:

||

y

-

Λ

⁢

(

x

1

+

x

2

+

…

+

x

n

)

||

<

ϵ

⁢

δ

⁢

2

-

n

(2)

Then by (1) we can pick

x

n

+

1

so that:

||

x

n

+

1

||

<

ϵ

⁢

2

-

n

(3)

and

||

y

-

Λ

⁢

(

x

1

+

x

2

+

…

+

x

n

)

-

Λ

⁢

(

x

n

+

1

)

||

<

ϵ

⁢

δ

⁢

2

-

(

n

+

1

)

, so (2) is satisfied for

x

n

+

1

.

Put

s

n

=

x

1

+

x

2

+

…

+

x

n

. Then from (3),

s

n

is a

Cauchy sequence

, and so, since

X

is

complete

, it converges to some

x

∈

X

. By (2),

Λ

⁢

s

n

→

y

, and by the

continuity

of

Λ

,

Λ

⁢

s

n

→

Λ

⁢

x

, so

Λ

⁢

x

=

y

. Also,

||

x

||

=

lim

n

→

∞

⁡

||

s

n

||

≤

∑

n

=

1

∞

||

x

n

||

<

1

+

ϵ

. Thus

Λ

⁢

(

(

1

+

ϵ

)

⁢

U

)

⊃

η

⁢

V

, or

Λ

⁢

(

U

)

⊃

(

1

+

ϵ

)

-

1

⁢

δ

⁢

V

. Since this is true for all

ϵ

>

0

, we have

Λ

⁢

(

U

)

⊃

∪

ϵ

>

0

(

1

+

ϵ

)

-

1

⁢

δ

⁢

V

=

δ

⁢

V

.

Title

proof of open mapping theorem

Canonical name

ProofOfOpenMappingTheorem

Date of creation

2013-03-22 16:23:31

Last modified on

2013-03-22 16:23:31

Owner

Statusx (15142)

Last modified by

Statusx (15142)

Numerical id

9

Author

Statusx (15142)

Entry type

Proof

Classification

msc 30A99

Classification

msc 46A30

Generated on Fri Feb 9 20:14:37 2018 by

LaTeXML
proof of Weierstrass M-test

proof of Weierstrass M-test

Consider the sequence of partial sums

s

n

=

∑

m

=

1

n

f

m

.
Take any

p

,

q

∈

ℕ

such that

p

≤

q

,then, for every

x

∈

X

, we have

|

s

q

⁢

(

x

)

-

s

p

⁢

(

x

)

|

=

|

∑

m

=

p

+

1

q

f

m

⁢

(

x

)

|

≤

∑

m

=

p

+

1

q

|

f

m

⁢

(

x

)

|

≤

∑

m

=

p

+

1

q

M

m

But since

∑

n

=

1

∞

M

n

converges

, for any

ϵ

>

0

we can
find an

N

∈

ℕ

such that, for any

p

,

q

>

N

and

x

∈

X

, we have

|

s

q

⁢

(

x

)

-

s

p

⁢

(

x

)

|

≤

∑

m

=

p

+

1

q

M

m

<

ϵ

. Hence the

sequence

s

n

converges uniformly

to

∑

n

=

1

∞

f

n

.

Title

proof of Weierstrass M-test

Canonical name

ProofOfWeierstrassMtest

Date of creation

2013-03-22 12:58:01

Last modified on

2013-03-22 12:58:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Proof

Classification

msc 30A99

Related topic

CauchySequence

Generated on Fri Feb 9 20:14:44 2018 by

LaTeXML