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Every homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and.

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## Some Related Sentences

Every and homomorphism

__Every__continuous map

**f**

**:**X

**→**Y

**induces**an algebra

__homomorphism__C (

**f**)

**:**C ( Y )

**→**C ( X ) by

**the**rule C (

**f**)( φ ) = φ o

**f**for every φ in C ( Y ).

__Every__vector v in determines

**a**linear map from R to taking 1 to v, which can be thought

**of**as

**a**

**Lie**algebra

__homomorphism__

**.**

__Every__completely multiplicative function is

**a**

__homomorphism__

**of**monoids

**and**is completely determined by its restriction to

**the**prime numbers

**.**

__Every__inner automorphism is indeed an automorphism

**of**

**the**group

**G**, i

**.**e

**.**it is

**a**bijective map from

**G**to

**G**

**and**it is

**a**

__homomorphism__; meaning ( xy )< sup >

**a**</ sup >

__Every__algebraic structure has its own notion

**of**

__homomorphism__, namely any function compatible with

**the**operation ( s ) defining

**the**structure

**.**

__Every__

__homomorphism__

**of**

**the**Petersen graph to itself that doesn't identify adjacent vertices is an automorphism

**.**

__Every__subgroup

**of**

**a**free abelian group is itself free abelian, which is important for

**the**description

**of**

**a**general abelian group as

**a**cokernel

**of**

**a**

__homomorphism__

**between**free abelian

**groups**

**.**

__Every__interior algebra

__homomorphism__is

**a**topomorphism, but not every topomorphism is an interior algebra

__homomorphism__

**.**

__Every__isogeny

**f**

**:**A

**→**B is automatically

**a**group

__homomorphism__

**between**

**the**

**groups**

**of**k-valued points

**of**A

**and**B, for any field k over which

**f**is defined

**.**

__Every__inverse semigroup S has

**a**E-unitary cover ; that is there exists an idempotent separating surjective

__homomorphism__from some E-unitary semigroup T onto S

**.**

Every and f

__Every__locally constant function from

**the**real numbers R to R is constant by

**the**connectedness

**of**R

**.**But

**the**function

__f__from

**the**rationals Q to R, defined by

__f__( x ) = 0 for x < π,

**and**

__f__( x ) = 1 for x > π, is locally constant ( here we use

**the**fact that π is irrational

**and**that therefore

**the**two sets

__Every__empty function is constant, vacuously, since there are no x

**and**y in A for which

__f__( x )

**and**

__f__( y ) are different when A is

**the**empty set

**.**

__Every__bounded positive-definite measure μ on

**G**satisfies μ ( 1 ) ≥ 0

**.**improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function

__f__on

**G**,

**the**function Δ < sup >– ½ </ sup >

__f__has non-negative integral with respect to Haar measure, where Δ denotes

**the**modular function

**.**

__Every__morphism

__f__

**:**

**G**

**→**

**H**in Grp has

**a**category-theoretic kernel ( given by

**the**ordinary kernel

**of**algebra ker

__f__=

__Every__solution

**of**

**the**second half g

**of**

**the**equation defines

**a**unique direction for x via

**the**first half

__f__

**of**

**the**equations, while

**the**direction for y is arbitrary

**.**

__Every__deterministic complexity class ( DSPACE (

__f__( n )), DTIME (

__f__( n )) for all

__f__( n )) is closed under complement, because one can simply add

**a**last step to

**the**algorithm which reverses

**the**answer

**.**

Every and G

Group actions / representations

**:**__Every__group__G__can be considered as**a**category with**a**single object whose morphisms are**the**elements**of**__G__**.**A functor from__G__to Set is then nothing but**a**group action**of**__G__on**a**particular set, i**.**e**.****a**G-set**.**__Every__open subgroup

**H**is also closed, since

**the**complement

**of**

**H**is

**the**open set given by

**the**union

**of**open sets gH for g in

__G__

*

__Every__connected graph__G__admits**a**spanning tree, which is**a**tree that contains every vertex**of**__G__**and**whose edges are edges**of**__G__**.**__Every__adjunction 〈 F,

__G__, ε, η 〉 gives rise to an associated monad 〈 T, η, μ 〉 in

**the**category D

**.**The functor

__Every__non-inner automorphism yields

**a**non-trivial element

**of**Out (

__G__), but different non-inner automorphisms may yield

**the**same element

**of**Out (

__G__).

__Every__one

**of**

**the**infinitely many vertices

**of**

__G__can be reached from v < sub > 1 </ sub > with

**a**simple path,

**and**each such path must start with one

**of**

**the**finitely many vertices adjacent to v < sub > 1 </ sub >.

__Every__smooth function

__G__over

**the**symplectic manifold generates

**a**one-parameter family

**of**symplectomorphisms

**and**if

__Every__year at

**the**International Astronautical Congress, three prestigious awards are given out

**:**

**the**Allan D

**.**Emil Memorial Award,

**the**Franck J

**.**Malina Astronautics Medal

**and**

**the**Luigi

__G__

**.**Napolitano Award

**.**

__Every__such regular cover is

**a**principal G-bundle, where

__G__= Aut ( p ) is considered as

**a**discrete topological group

**.**

*

__Every__Polish space is homeomorphic to**a**__G__< sub > δ </ sub > subspace**of****the**Hilbert cube,**and**every__G__< sub > δ </ sub > subspace**of****the**Hilbert cube is Polish**.**

Every and →

*

__Every__separable metric space is isometric to**a**subset**of**C (),**the**separable Banach space**of**continuous functions__→__R, with**the**supremum norm**.**__Every__smooth ( or differentiable ) map φ

**:**M

__→__N

**between**smooth ( or differentiable ) manifolds

**induces**natural linear maps

**between**

**the**

**corresponding**tangent spaces

**:**

__Every__universal cover p

**:**D

__→__X is regular, with deck transformation group being isomorphic to

**the**fundamental group

**.**

__Every__Boolean algebra is

**a**Heyting algebra when

**a**

__→__b is defined as usual as ¬

**a**∨ b, as is every complete distributive lattice when

**a**

__→__b is taken to be

**the**supremum

**of**

**the**set

**of**all c for which

**a**∧ c ≤ b

**.**The open sets

**of**

**a**topological space form

**a**complete distributive lattice

**and**hence

**a**Heyting algebra

**.**

__Every__functor F

**:**D

__→__E

**induces**

**a**functor F < sup > C </ sup >

**:**D < sup > C </ sup >

__→__E < sup > C </ sup > ( by composition with F ).

0.676 seconds.