espaces vectoriels

K K
⊕
∀~,~,~∈
~⊕~∈
(~⊕~)⊕~=~⊕(~⊕~)
~⊕~=~⊕~
~∀~∈~⊕=~
−~
∀~∈~⊕(−~) = = ~
K
• ∀~,~,~∈,∀λ, µ ∈K
λ•~∈
λ•(µ•~)=(λ×µ)•~
(λ+µ)•~=λ•~⊕µ•~•
λ•(~⊕~) = λ•~⊕λ•~•
•~=~•
(R,+,·)R
(R[ ],+,×)
R
(C[ ],+,×)
R C
F(,K) = {:→K}K
·
λ·~λ•~+⊕
(,+,·)K∀~∈,∀λ, µ ∈K
·~=
λ·=
(−)·~=−~
(λ−µ)·~=λ·~−µ·~
λ·~=⇐⇒ λ=~=
(,+,·)K
∀~,~∈,~+~∈
∀~∈,∀λ∈K, λ~∈
(,+,·)
(,+,·)
⊂ ⇐⇒ (∈
∀λ∈K,∀~,~∈, λ~+~∈
K[ ] = { ∈ K[ ],( ) ≤ } K[ ]
={ ∈ K[ ],( ) = }
K[ ]
∈ − − ∈ + (−+ ) = /∈
C(R,R)F(R,R)
P={(, , )∈R,+ + = }
R
~∈
K·~={ · ~,∈K}
~
~

K∩
( ) =.. T
=
R·~∪R·~~+~
R R
K
{~,~· · · ,~}K
λ , λ , · · · , λ K
λ~+λ~+· · · +λ~
~,· · · ,~
V={~,~· · · ,~}
V
(V)
(V) = (X
=
λ~, λ ∈K,∈ { ,· · · ,})
(V)V
( ) ⊂
( )
~6=~K·~= (~)
~
~ ~
(~,~)
(~,~) = {λ~+µ~, λ ∈K, µ ∈K}
K[ ] ≤
,
({, , }) = {+ + , , , ∈K}=K[ ]
KV={, , . . . , }
V
λ+λ+. . . +λ=⇒λ=λ=. . . =λ=.
Vλ , . . . , λ
λ+λ+. . . +λ=
V V
V
R
~= ( ,)~= ( ,){ ~,~}
λ
~+µ ~=~⇒(λ, µ) = ( ,)⇒λ=µ=
~= ( ,)~= ( ,){~,~}
~=~ ~ −~=~
R~= ( ,,)~= ( , , −)
~= ( ,,){~,~,~}
α~+β~+γ~=~⇐⇒ 
α+γ=
β=
α−β+γ=
α=β=γ=
R{(,−,)
| {z }
,(−, , −)
| {z }
,(,−,−)
| {z }
}
− +
−
−
+ −
−
= ⇐⇒
−+ =
−+−=
− − =
⇐⇒ · · · ⇐⇒ (=
=
= + + = ~
K
F(R,R){,}
λ+µ=˜⇐⇒ ∀ ∈ R, λ ( ) + µ( ) = .
=
λ= = πµ=
F(R,R){, , }
+ =
K{,..., }
= ( ,..., )
,...,
∀ ∈ ,∃λ , . . . , λ ∈K,=λ+. . . +λ .
{
~,~,~}R(, , )∈R
(, , ) = ( ,,) + ( , , ) + ( ,,) = ~+~+~
{, , , . . . , }R[ ]
( ( ) = + + + . . . + )
{, , , , . . . , }
( ( ) = + + ×+ + . . . + )
→
K
RB= ,· · · ,
R
{, , , , . . . , }
R[ ] +

B={,..., }
,...,
∀ ∈ ,∃!(α , . . . , α )∈K,=X
=
α .
α , . . . , α B
[ ]B=
α
α
R[ ] = −[ ]B=
−
B={, , }
RB0={(,),(−,)}
 = +−⇐⇒ (=−
= + ⇐⇒
· · · ⇐⇒ (=+
=−
B0
 B0
=+
−
K
R R [ ] ≤
R[ ] C[ ] F(R,R)
6={ } K
K
( ) =
({ }) =
R=R=R=
R[ ] +
+
B={(, , ),(,,),(,,)}
R(,−,)
B={,(−),(−),(−)}
R[ ]
={(, , , )∈R,+ + + = + −=}
R
=C∞(R,R)
={ ∈ ,+0+00 =}
K
( ) ≤( )
( ) = ( ) =
⊂ ≤
⊂= =
R
{(, , )}
(~),~6=~
(~,~),~ ~
R
F={,..., }K
FF(F)
(F) = ( (F))
({,..., }) = ⇐⇒ { ,..., }
F={,..., }(F)
(F)F

RF={(,,),(,,),(,−,)}
R[ ] P={− , , +,− }
KB={,..., }
{,..., }
B,..., B
. . . +. . .
?
?
?
?
. . .
?
?
. . .
?
F
F(F)
R({(,,,−),(−,,,),(, , , −)}
(F)
R[ ] = + −+,=−+
= + + −= + −
(,,,) ( ,,,)
={+,∈,∈ }
= +
+∩={ }
⊕
+ = ( ∪) +
+
+
R=R
~⊕R ~= (( ,)) ⊕(( ,))
R
P={(, , )∈R,=}={(, , ),∈R,∈R}
P={(, , )∈R,=}={(, , ),∈R,∈R}
R=P+P
=F(R,R)P={:R→R,}
I={:R→R,}=P ⊕ I
R
K
=⊕
+ = ∩={ }
⊕=⊕;=
K
( + ) = ( ) + ( ) −(∩).
(⊕) = ( ) + ( ).
B B
B ∪ B ⊕
B ∪ B ⊕ =
( ) =
−
={(, , )∈R,+−=}
={(, , −),∈R} ⊕ =R
R
V= ({, , })
= ( , , −, , −),= ( ,,,,) = ( ,,,,)