aarjaneiro/planetmath huggingface dataset:
successor cardinal

successor cardinal

A

successor cardinal

is a

cardinal

that is the

cardinal successor

of some cardinal.

Every finite cardinal other than

0

is a successor cardinal.
An

infinite

cardinal

ℵ

α

is a successor cardinal if and only if

α

is a

successor ordinal

.

Title

successor cardinal

Canonical name

SuccessorCardinal

Date of creation

2013-03-22 14:04:47

Last modified on

2013-03-22 14:04:47

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Related topic

LimitCardinal

Related topic

CardinalSuccessor

Generated on Thu Feb 8 20:11:26 2018 by

LaTeXML
the Cartesian product of a finite number of countable sets is countable

the Cartesian product of a finite number of countable sets is countable

Theorem 1

The

Cartesian product

of a finite number of

countable sets

is

countable

.

Proof:

Let

A

1

,

A

2

,

…

,

A

n

be countable sets and let

S

=

A

1

×

A

2

×

⋯

×

A

n

.
Since each

A

i

is countable, there exists an

injective function

f

i

:

A

i

→

ℕ

.
The

function

h

:

S

→

ℕ

defined by

h

⁢

(

a

1

,

a

2

,

…

⁢

a

n

)

=

∏

i

=

1

n

p

i

f

i

⁢

(

a

i

)

where

p

i

is the

i

th prime
is, by the

fundamental theorem of arithmetic

, a

bijection

between

S

and a subset of

ℕ

and therefore

S

is also countable.

Note that this result does

not

(in general) extend to the
Cartesian product of a

countably infinite

collection

of countable
sets. If such a collection contains more than a finite number of sets
with at least two

elements

, then Cantor’s diagonal argument can be
used to show that the

product

is not countable.

For example, given

B

=

{

0

,

1

}

, the set

F

=

B

×

B

×

⋯

consists of all countably infinite

sequences

of zeros and ones.
Each element of

F

can be viewed as a binary

fraction

and can
therefore be mapped to a unique
real number in

[

0

,

1

)

and

[

0

,

1

)

is, of course, not countable.

Title

the Cartesian product of a finite number of countable sets is countable

Canonical name

TheCartesianProductOfAFiniteNumberOfCountableSetsIsCountable

Date of creation

2013-03-22 15:19:45

Last modified on

2013-03-22 15:19:45

Owner

BenB (9643)

Last modified by

BenB (9643)

Numerical id

13

Author

BenB (9643)

Entry type

Theorem

Classification

msc 03E10

Synonym

The product of a finite number of countable sets is countable

Related topic

CardinalityOfACountableUnion

Related topic

AlgebraicNumbersAreCountable

Related topic

CardinalityOfTheRationals

Generated on Thu Feb 8 20:11:28 2018 by

LaTeXML
the set of all real transcendental numbers is uncountable

the set of all real transcendental numbers is uncountable

Theorem

.

The

set of all real transcendental numbers is uncountable

.

Proof.

Denote

𝕋

and

𝔸

be the set of real transcendental and real

algebraic numbers

respectively. Suppose

𝕋

is

countable

. Then the union

𝕋

∪

𝔸

=

ℝ

is also countable, since

𝔸

is also countable, which is a

contradiction

. Therefore

𝕋

must be

uncountable

.

□

Title

the set of all real transcendental numbers is uncountable

Canonical name

TheSetOfAllRealTranscendentalNumbersIsUncountable

Date of creation

2013-03-22 16:08:05

Last modified on

2013-03-22 16:08:05

Owner

gilbert_51126 (14238)

Last modified by

gilbert_51126 (14238)

Numerical id

11

Author

gilbert_51126 (14238)

Entry type

Theorem

Classification

msc 03E10

Generated on Thu Feb 8 20:11:30 2018 by

LaTeXML
uncountable

uncountable

Definition

A set is

uncountable

if it is not

countable

.
In other words, a set

S

is uncountable, if
there is no subset of

ℕ

(the set of natural numbers) with the same

cardinality

as

S

.

1.

All uncountable sets are

infinite

. However, the

converse

is not
true, as

ℕ

is both infinite and countable.

2.

The real numbers form an uncountable set. The famous proof
of this result is based on Cantor’s diagonal argument.

Title

uncountable

Canonical

name

Uncountable

Date of creation

2013-03-22 11:59:08

Last modified on

2013-03-22 11:59:08

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Synonym

uncountable set

Related topic

CardinalityOfTheContinuum

Generated on Thu Feb 8 20:11:32 2018 by

LaTeXML
Veblen function

Veblen function

The

Veblen function

is used to obtain larger

ordinal numbers

than those provided by

exponentiation

. It builds on a hierarchy of closed and

unbounded

classes:

•

C

⁢

r

⁢

(

0

)

is the

additively indecomposable

numbers,

ℍ

•

C

⁢

r

⁢

(

S

⁢

n

)

=

C

⁢

r

⁢

(

n

)

′

the set of

fixed points

of the enumerating function of

C

⁢

r

⁢

(

n

)

•

C

⁢

r

⁢

(

λ

)

=

⋂

α

<

λ

C

⁢

r

⁢

(

α

)

The Veblen function

φ

α

⁢

β

is defined by setting

φ

α

equal to the enumerating function of

C

⁢

r

⁢

(

α

)

.

We call a number

α

strongly critical

if

α

∈

C

⁢

r

⁢

(

α

)

. The class of strongly critical

ordinals

is written

𝐒𝐂

, and the enumerating function is written

f

𝐒𝐂

⁢

(

α

)

=

Γ

α

.

Γ

0

, the first strongly critical ordinal, is also called the

Feferman-Schutte

ordinal.

Title

Veblen function

Canonical name

VeblenFunction

Date of creation

2013-03-22 13:29:10

Last modified on

2013-03-22 13:29:10

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Classification

msc 03F15

Defines

strongly critical

Defines

Feferman-Schutte ordinal

Generated on Thu Feb 8 20:11:46 2018 by

LaTeXML
von Neumann integer

von Neumann integer

A

von Neumann

is not an integer, but instead a construction of a

natural number

using some basic set notation. The

von Neumann integers

are defined inductively. The von Neumann integer zero is defined to be the

empty set

,

∅

, and there are no smaller von Neumann integers.
The von Neumann integer

N

is then the set of all von Neumann integers less than

N

. The set of von Neumann integers is the set of all finite

von Neumann ordinals

(

http://planetmath.org/VonNeumannOrdinal

).

This form of construction from very basic notions of sets is applicable to various forms of

set theory

(for

instance

,

Zermelo-Fraenkel set theory

). While this construction suffices to define the set of natural numbers, a little more work must be done to define the set of all integers (

http://planetmath.org/Integer

).

Examples

0

=

∅

1

=

{

0

}

=

{

∅

}

2

=

{

0

,

1

}

=

{

∅

,

{

∅

}

}

3

=

{

0

,

1

,

2

}

=

{

∅

,

{

∅

}

,

{

{

∅

,

{

∅

}

}

}

}

⋮

N

=

{

0

,

1

,

…

,

N

-

1

}

Title

von Neumann integer

Canonical name

VonNeumannInteger

Date of creation

2013-03-22 12:32:34

Last modified on

2013-03-22 12:32:34

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E10

Related topic

NaturalNumber

Related topic

VonNeumannOrdinal

Generated on Thu Feb 8 20:11:48 2018 by

LaTeXML
von Neumann ordinal

von Neumann ordinal

The

is a method of defining

ordinals

in

set theory

.

The

von Neumann ordinal

α

is defined to be the

well-ordered set

containing the von Neumann ordinals which precede

α

. The set of finite von Neumann ordinals is known as the

von Neumann integers

. Every well-ordered set is

isomorphic

to a von Neumann ordinal.

They can be constructed by

transfinite recursion

as follows:

•

The

empty set

is

0

.

•

Given any ordinal

α

, the ordinal

α

+

1

(the

successor

of

α

) is defined to be

α

∪

{

α

}

.

•

Given a set

A

of ordinals,

⋃

a

∈

A

a

is an ordinal.

If an ordinal is the successor of another ordinal, it is an

successor ordinal

. If an ordinal is neither

0

nor a successor ordinal then it is a

limit ordinal

. The first limit ordinal is named

ω

.

The class of ordinals is denoted

𝐎𝐧

.

The von Neumann ordinals have the convenient property that if

a

<

b

then

a

∈

b

and

a

⊂

b

.

Title

von Neumann ordinal

Canonical name

VonNeumannOrdinal

Date of creation

2013-03-22 12:32:37

Last modified on

2013-03-22 12:32:37

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

11

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Synonym

ordinal

Related topic

VonNeumannInteger

Related topic

ZermeloFraenkelAxioms

Related topic

OrdinalNumber

Defines

successor ordinal

Defines

limit ordinal

Defines

successor

Generated on Thu Feb 8 20:11:50 2018 by

LaTeXML
weakly compact cardinal

weakly compact cardinal

Weakly compact cardinals

are (large)

infinite

cardinals

which have a

property

related to the

syntactic compactness theorem for first order logic

. Specifically, for any infinite cardinal

κ

, consider the

language

L

κ

,

κ

.

This language is identical to first logic except that:

•

infinite

conjunctions

and

disjunctions

of fewer than

κ

formulas

are allowed

•

infinite strings of fewer than

κ

quantifiers

are allowed

The weak compactness theorem for

L

κ

,

κ

states that if

Δ

is a set of

sentences

of

L

κ

,

κ

such that

|

Δ

|

=

κ

and any

θ

⊂

Δ

with

|

θ

|

<

κ

is

consistent

then

Δ

is consistent.

A cardinal is weakly compact if the weak compactness theorem holds for

L

κ

,

κ

.

Title

weakly compact cardinal

Canonical

name

WeaklyCompactCardinal

Date of creation

2013-03-22 12:50:53

Last modified on

2013-03-22 12:50:53

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Synonym

weakly compact

Related topic

CardinalNumber

Defines

weakly compact cardinal

Defines

weak compactness theorem

Generated on Thu Feb 8 20:11:52 2018 by

LaTeXML
weakly compact cardinals and the tree property

weakly compact cardinals and the tree property

A

cardinal

is

weakly compact

if and only if it is

inaccessible

and has the

tree property

.

Weak

compactness

implies tree property

Let

κ

be a weakly compact cardinal and let

(

T

,

<

T

)

be a

κ

tree with all levels smaller than

κ

. We define a theory in

L

κ

,

κ

with for each

x

∈

T

, a

constant

c

x

, and a single

unary relation

B

. Then our theory

Δ

consists of the

sentences

:

•

¬

⁡

[

B

⁢

(

c

x

)

∧

B

⁢

(

c

y

)

]

for every incompatible

x

,

y

∈

T

•

⋁

x

∈

T

⁢

(

α

)

B

⁢

(

c

x

)

for each

α

<

κ

It should be clear that

B

represents membership in a

cofinal branch

, since the first class of sentences asserts that no incompatible

elements

are both in

B

while the second class states that the branch

intersects

every level.

Clearly

|

Δ

|

=

κ

, since there are

κ

elements in

T

, and hence fewer than

κ

⋅

κ

=

κ

sentences in the first group, and of course there are

κ

levels and therefore

κ

sentences in the second group.

Now consider any

Σ

⊆

Δ

with

|

Σ

|

<

κ

. Fewer than

κ

sentences of the second group are included, so the set of

x

for which the corresponding

c

x

must all appear in

T

⁢

(

α

)

for some

α

<

κ

. But since

T

has branches of arbitrary height,

T

⁢

(

α

)

⊧

Σ

.

Since

κ

is weakly compact, it follows that

Δ

also has a model, and that model obviously has a set of

c

x

such that

B

⁢

(

c

x

)

whose corresponding elements of

T

intersect every level and are

compatible

, therefore forming a cofinal branch of

T

, proving that

T

is not Aronszajn.

Title

weakly compact cardinals and the tree property

Canonical

name

WeaklyCompactCardinalsAndTheTreeProperty

Date of creation

2013-03-22 12:52:51

Last modified on

2013-03-22 12:52:51

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Result

Classification

msc 03E10

Related topic

TreeProperty

Related topic

Aronszajn

Generated on Thu Feb 8 20:11:54 2018 by

LaTeXML
well-founded recursion

well-founded recursion

Theorem 1

.

Let

G

be a binary (class)

function

on

V

, the class of all sets. Let

A

be a

well-founded

set (with

R

the well-founded relation). Then there is a unique function

F

such that

F

(

x

)

=

G

(

x

,

F

|

seg

(

x

)

)

,

where

seg

⁡

(

x

)

:=

{

y

∈

A

∣

y

⁢

R

⁢

x

}

, the

initial segment

of

x

.

Remark

. Since every

well-ordered set

is well-founded, the

well-founded recursion

theorem

is a

generalization

of the

transfinite recursion

theorem. Notice that the

G

here is a function in two

arguments

, and that it is necessary to specify the

element

x

in the first argument (in contrast with the

G

in the transfinite recursion theorem), since it is possible that

seg

⁡

(

a

)

=

seg

⁡

(

b

)

for

a

≠

b

in a well-founded set.

By converting

G

into a

formula

(

φ

⁢

(

x

,

y

,

z

)

such that for all

x

,

y

, there is a unique

z

such that

φ

⁢

(

x

,

y

,

z

)

), then the above theorem can be proved in ZF (with the aid of the

well-founded induction

). The proof is

similar

to the proof of the transfinite recursion theorem.

Title

well-founded recursion

Canonical name

WellfoundedRecursion

Date of creation

2013-03-22 17:54:52

Last modified on

2013-03-22 17:54:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E10

Classification

msc 03E45

Related topic

TransfiniteRecursion

Generated on Thu Feb 8 20:12:04 2018 by

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κ-complete

κ

-complete

A structured set

S

(typically a filter or a

Boolean algebra

) is

κ

-

complete

if, given any

K

⊆

S

with

|

K

|

<

κ

,

⋂

K

∈

S

. It is

complete

if it is

κ

-complete for all

κ

.

Similarly, a

partial order

is

κ

-complete if any

sequence

of fewer than

κ

elements

has an

upper bound

within the partial order.

A

ℵ

1

-complete is called

countably complete

.

Title

κ

-complete

Canonical name

kappacomplete

Date of creation

2013-03-22 12:53:07

Last modified on

2013-03-22 12:53:07

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

11

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Synonym

kappa-complete

Synonym

kappa complete

Related topic

Filter

Related topic

BooleanAlgebra

Defines

countably complete

Generated on Thu Feb 8 20:12:06 2018 by

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Cantor-Bendixson theorem

Cantor-Bendixson theorem

Any

closed subset

X

of the reals can be written as a

disjoint union

X

=

C

∪

P

,

where

C

is

countable

and

P

is a

perfect set

(hence this

theorem

is also known as the

CUP theorem

).

Title

Cantor-Bendixson theorem

Canonical name

CantorBendixsonTheorem

Date of creation

2013-03-22 15:01:40

Last modified on

2013-03-22 15:01:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E15

Classification

msc 54H05

Synonym

CUP theorem

Generated on Thu Feb 8 20:12:08 2018 by

LaTeXML
Cantor’s theorem

Cantor’s theorem

Let

X

be any set and

𝒫

⁢

(

X

)

its

power set

. Then there is no

bijection

between

X

and

𝒫

⁢

(

X

)

. Moreover, the

cardinality

of

𝒫

⁢

(

X

)

is strictly greater than that of

X

; that is,

|

X

|

<

|

𝒫

⁢

(

X

)

|

.

Title

Cantor’s

theorem

Canonical name

CantorsTheorem

Date of creation

2013-03-22 12:44:50

Last modified on

2013-03-22 12:44:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 03E17

Classification

msc 03E10

Related topic

CantorsDiagonalArgument

Related topic

KonigsTheorem

Generated on Thu Feb 8 20:12:09 2018 by

LaTeXML
cardinality of the continuum

cardinality of the continuum

The

cardinality

of the

continuum

, often denoted by

𝔠

, is
the cardinality of the set

ℝ

of real numbers.
A set of cardinality

𝔠

is said to have

continuum many

elements

.

Cantor’s diagonal argument shows that

𝔠

is

uncountable

.
Furthermore, it can be shown that

ℝ

is equinumerous with the

power set

of

ℕ

, so

𝔠

=

2

ℵ

0

.
It can also be shown that

𝔠

has uncountable

cofinality

.

It can also be shown that

𝔠

=

𝔠

ℵ

0

=

ℵ

0

⁢

𝔠

=

𝔠

⁢

𝔠

=

𝔠

+

κ

=

𝔠

n

for all finite

cardinals

n

≥

1

and all cardinals

κ

≤

𝔠

.
See the article on

cardinal arithmetic

for some of the basic facts underlying these equalities.

There are many

properties

of

𝔠

that

independent

of ZFC,
that is, they can neither be proved nor disproved in ZFC,
assuming that ZF is

consistent

.
For example, for every nonzero

natural number

n

,
the equality

𝔠

=

ℵ

n

is independent of ZFC.
(The case

n

=

1

is the well-known

Continuum Hypothesis

(

http://planetmath.org/ContinuumHypothesis

).)
The same is true for most other alephs,
although in some cases equality can be ruled out on the grounds of cofinality,
e.g.,

𝔠

≠

ℵ

ω

.
In particular,

𝔠

could be either

ℵ

1

or

ℵ

ω

1

,
so it could be either a

successor cardinal

or a

limit cardinal

,
and either a regular cardinal or a singular cardinal.

Title

cardinality of the continuum

Canonical

name

CardinalityOfTheContinuum

Date of creation

2013-03-22 14:15:33

Last modified on

2013-03-22 14:15:33

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

19

Author

yark (2760)

Entry type

Definition

Classification

msc 03E17

Classification

msc 03E10

Synonym

cardinal of the continuum

Synonym

cardinal number

of the continuum

Related topic

CardinalNumber

Related topic

CardinalArithmetic

Defines

continuum many

Generated on Thu Feb 8 20:12:11 2018 by

LaTeXML
proof of Cantor’s theorem

proof of Cantor’s theorem

The proof of this

theorem

is fairly using the following construction, which is

central

to Cantor’s diagonal argument.

Consider a

function

F

:

X

→

𝒫

⁢

(

X

)

from a set

X

to its

power set

. Then we define the set

Z

⊆

X

as follows:

Z

=

{

x

∈

X

∣

x

∉

F

⁢

(

x

)

}

Suppose that

F

is a

bijection

. Then there must exist an

x

∈

X

such that

F

⁢

(

x

)

=

Z

. Then we have the following

contradiction

:

x

∈

Z

⇔

x

∉

F

⁢

(

x

)

⇔

x

∉

Z

Hence,

F

cannot be a bijection between

X

and

𝒫

⁢

(

X

)

.

Title

proof of Cantor’s theorem

Canonical name

ProofOfCantorsTheorem

Date of creation

2013-03-22 12:44:55

Last modified on

2013-03-22 12:44:55

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 03E17

Classification

msc 03E10

Generated on Thu Feb 8 20:12:13 2018 by

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additive

additive

Let

ϕ

be some positive-valued

set function

defined on an

algebra of sets

𝒜

. We say that

ϕ

is

additive

if, whenever

A

and

B

are

disjoint sets

in

𝒜

, we have

ϕ

⁢

(

A

∪

B

)

=

ϕ

⁢

(

A

)

+

ϕ

⁢

(

B

)

.

Given any

sequence

⟨

A

i

⟩

of disjoint sets in A and whose union is also in A, if we have

ϕ

⁢

(

⋃

A

i

)

=

∑

ϕ

⁢

(

A

i

)

we say that

ϕ

is

countably additive

or

σ

-additive

.

Useful properties of an additive set function

ϕ

include the following:

1.

ϕ

⁢

(

∅

)

=

0

.

2.

If

A

⊆

B

, then

ϕ

⁢

(

A

)

≤

ϕ

⁢

(

B

)

.

3.

If

A

⊆

B

, then

ϕ

⁢

(

B

∖

A

)

=

ϕ

⁢

(

B

)

-

ϕ

⁢

(

A

)

.

4.

Given

A

and

B

,

ϕ

⁢

(

A

∪

B

)

+

ϕ

⁢

(

A

∩

B

)

=

ϕ

⁢

(

A

)

+

ϕ

⁢

(

B

)

.

Title

additive

Canonical name

Additive

Date of creation

2013-03-22 13:00:58

Last modified on

2013-03-22 13:00:58

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

10

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 03E20

Synonym

additivity

Defines

countable additivity

Defines

countably additive

Defines

σ

-additive

Defines

sigma-additive

Generated on Thu Feb 8 20:12:15 2018 by

LaTeXML
antisymmetric

antisymmetric

A

relation

ℛ

on

A

is

antisymmetric

iff

∀

x

,

y

∈

A

,

(

x

ℛ

y

∧

y

ℛ

x

)

→

(

x

=

y

)

.
For a

finite set

A

with

n

elements

, the number of possible

antisymmetric relations

is

2

n

⁢

3

n

2

-

n

2

out of the

2

n

2

total possible
relations.

Antisymmetric is not the same thing as “not

symmetric

”, as it is possible
to have both at the same time. However, a relation

ℛ

that is both
antisymmetric and symmetric has the condition that

x

⁢

ℛ

⁢

y

⇒

x

=

y

.
There are only

2

n

such possible relations on

A

.

An example of an antisymmetric relation on

A

=

{

∘

,

×

,

⋆

}

would be

ℛ

=

{

(

⋆

,

⋆

)

,

(

×

,

∘

)

,

(

∘

,

⋆

)

,

(

⋆

,

×

)

}

.
One relation that isn’t antisymmetric is

ℛ

=

{

(

×

,

∘

)

,

(

⋆

,

∘

)

,

(

∘

,

⋆

)

}

because we have both

⋆

ℛ

∘

and

∘

ℛ

⋆

, but

∘

⁣

≠

⁣

⋆

Title

antisymmetric

Canonical name

Antisymmetric

Date of creation

2013-03-22 12:15:50

Last modified on

2013-03-22 12:15:50

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

14

Author

aoh45 (5079)

Entry type

Definition

Classification

msc 03E20

Synonym

antisymmetry

Related topic

Reflexive

Related topic

Symmetric

Related topic

ExteriorAlgebra

Related topic

SkewSymmetricMatrix

Generated on Thu Feb 8 20:12:17 2018 by

LaTeXML
argument

argument

The

argument

of a function is its input. For example, in the

expression

f

⁢

(

x

)

,

x

is the argument of

f

.

A common error for those who are unfamiliar with mathematics is to treat a function and its argument as two separate entities. For example, in solving the equation

ln

⁡

x

=

5

for

x

, people who are unfamiliar with mathematics may give the erroneous answer

x

=

5

ln

. This error might be circumvented by stressing that a function and its argument are not multiplied, but rather that a function its argument.

Another common error is to try to separate the argument of a function. This error is most common when the argument consists of at least two . For example, students may write

f

⁢

(

x

+

5

)

=

f

⁢

(

x

)

+

f

⁢

(

5

)

regardless of what the function

f

is.

Title

argument

Canonical

name

Argument

Date of creation

2013-03-22 16:07:06

Last modified on

2013-03-22 16:07:06

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03E20

Classification

msc 97D70

Generated on Thu Feb 8 20:12:19 2018 by

LaTeXML
characterization of basis of finite dimensional vector space

characterization of basis of finite dimensional vector space

Let

X

be a

linear space

and let

ϕ

i

be the

linear functional

,

ϕ

i

:

X

→

ℝ

,

1

≤

i

≤

n

, such as

[

ϕ

i

(

υ

)

=

0

,

∀

i

=

1

,

2

,

…

,

n

]

→

[

ϕ

(

υ

)

=

0

]

.

Then there exist

λ

1

,

λ

2

,

…

,

λ

n

∈

ℝ

such as

ϕ

=

∑

i

=

1

n

λ

i

⁢

ϕ

i

.

Title

characterization of basis of finite dimensional vector space

Canonical name

CharacterizationOfBasisOfFiniteDimensionalVectorSpace

Date of creation

2013-03-22 15:24:28

Last modified on

2013-03-22 15:24:28

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

9

Author

georgiosl (7242)

Entry type

Corollary

Classification

msc 03E20

Generated on Thu Feb 8 20:12:21 2018 by

LaTeXML