Chapitre 14 :Produit scalaire sur un R-ev
Transcription
Chapitre 14 :Produit scalaire sur un R-ev
T his work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license. ! "# $ ) * % ϕ • ,. • • - ϕ, ∈ λ∈ )∈ λ ∈ ) + # × -2 3 ϕ ϕ, + ,& 1 1+ ' ,, × !!! !!! + , ' # ( + ϕ, + ϕ , + λ ) + = ϕ , + + λϕ , ) + ϕ, + λ ) + = ϕ , + + λϕ , ) + , + = ϕ, ∈ ϕ , + ≥ / ,0 ( +=/ = / ,0 # ∈ & ' → → + + ! ϕ ! ! 4 ++ = % ' " 5 ! & ϕ, += 6 ! ≥/ = +7 &(3 8 9 7 3 , ϕ &(3 8 9+ ! ∈ (ϕ , " 3 ∈ • 3 =/ ϕ, + ϕ, • 3 ≠ /! λ∈ 6 ϕ ,λ + λ + + = λ ϕ , + + ϕ , ≥/ +) ≤ ϕ , 6 + ×ϕ, + , + + λϕ , + ϕ + + λϕ , # + ϕ = λ6ϕ , + +ϕ, + + 6λϕ , + Ismaël Bouya http://melusine.eu.org/syracuse/immae/ + 4 # ( ≠/ ϕ &: λ ' #, +! " ∆ = ϕ, +6 − ϕ, +ϕ , "% < % ! .+ = + ≠ /! 4 #! 3 ; * +≤/ * "# $ •∀ ∈ •∀ ∈ % % , +≥/ , , +=/ •∀ ∈ → , + +≤ , + , ++ , +, ϕ 3 + ! ∈ , " " ;* ' ∈ ϕ, ' = /+ ,λ + = λ •∀ ∈ > ϕ, ! " ϕ, +=/ ϕ, % , + = ϕ, 6 + + ϕ, + ≥ /! " ' % + + ' = /! ' λ∈ ϕ ,λ λ + = λ ϕ , + = λ ϕ , + ! ∈ !4 ' 3 , + + ≤ , + + , + ⇔ ϕ, + + + ≤ ϕ, + + ϕ, + ∈ 6 ⇔ ϕ, + ⇔ ϕ, + + ≤ ϕ, + + 6ϕ , + + 6 ϕ, + + ϕ, +ϕ , + ≤ ϕ, ' ϕ, + ≤ ϕ, +ϕ , + + ≤ ϕ, + ≤ ϕ, +ϕ , + , "# $ •∀ •∀ •∀ •∀ +ϕ , + + ϕ, + &(3 8 9+ ϕ 4 +" + + + 6 ϕ, ⇔ ϕ, + + ϕ, * % ∈ , +≥/ ∈ , , +=/ = + ∈ , += , + ∈ , +≤ , ++ , × → ' + 6 3 % , * " 4 •• , += , − + , − +≥/ % , += , − + + ! = ! − =/ ' • , += , − += ,−, − ++ = − , − += , • , += , − += , − + − +≤ , − ++ 3 + , − +≤ , ++ , ϕ * * + ϕ! % "+ 4 4 "# 3 ' ϕ, 0> ϕ! % ∈ !4 + = /! 4 ' ,⊥ ⊥ ϕ+ 2, & ( ( * 1+! ' ⊥/ ∀ ∈ ϕ, / + = / ! , & + ∀ ∈ 7 3 ⊥ ⇔ ∈ !4 %' " 3 , + + = , + + = ϕ, + = , + + 6ϕ , 6 , + + = 6 6 + + = ϕ, 6 , + + 6 ϕ! , + + 6 ++ , + + + 6ϕ , , + ∈ 6 + + ϕ, + +=/⇔ ⊥ 6 , + ⇔ ϕ, 6 ! -+ " 4 " ϕ % , + +6 = , + 6 + 6ϕ , ++ , + 6 ,- , − + = , + − 6ϕ , ++ 6 , + +6 + , − +6 = 6 , +6 + 6 , +6 , 6 6 , + + − , − + = ϕ, 6 6 , + ! @ + + + ? A+ A 4 "# 3 , +∈ • 5 # ϕ! % # ! ∈ , ≠ ⇔∀ # • 5 # ∀ ⇔ # ∈ , ≠ ∀ ∈ = = / + ⊥ ∈ ϕ, ⇔∀ ,4< δ = 5 # ⊥ + + =δ + , +∈ ⇔ ⊂ # ⇔ ⊂ # # λ # , +∈ ! ,λ + ∈ # =/ λ =/ ∀ ∈ ∈ 3 , +∈ # , # , +∈ " 3 + , +∈ ,λ + ∈ ! , +∈ # ⊂ !3 # !3 λ ' # !3 =/ ! ∈ ϕ, ϕ, ∈ ! 3 λ += ∈ λ ϕ, ∈ λ = /! " ∀ ∈ " -2 /+ = / =/ , +∈ + = λ ϕ, ≠ λ = /! +!" ≠/ !" , +∈ ! # 5 ' ! = / ,C/ 6π B , + ∈ # ∈ 6π / , + , + = 6π 6 # ,, + + + + / 6π + / ! # !- ,, − + + 6π ≠ = 6 + ,, + + + + − ,, − + + 6π = ≠/ = =/ = = 6 6 + [6 ]/6π =π ≠ / ,, + + + + / = 6π ≠ / =/ / ## , # ∈ π E+ 3 + D 2 4 "# 3 ϕ! % ⊥ ! 4 , 2* ⊥ ={ ∈ +! % " • / ∈ ⊥ ∀ ∈ ϕ ,/ • 3 )∈ ⊥ λ ∈ ! + λ )∈ ⊥ ∀ ∈ , ' % ⊥ "# 3 ∀ ∈ ⊥ ! +=/ ϕ , + λ ) + = ϕ , + + λϕ , ) + = / + λ / = / ! # ' ' > ' ' F+ ( 2 2⇔∀ ∈ # ⊥ 4 -2 , ⊥ ' • {/ } = ⇔ ⊂ ⊥ ⇔ ⊥ ⊂ ! ? + )⊂ ⊥ ! ⊥ ∀ ∈ ⊥ ∀ ∈ • = {/ }! - ## 3 ∀ ∈ ∈ ⊥! ⊥ "% < % ⊂ {/ } ϕ, / + = / ⊥ •, 05 ϕ, ϕ, + = /! - + = /! " = /! F ⊥ ⊥ + ⊃ 2* 2* 4 3 ∈ ' ∈ 5% # # ' 1! ∈, ∈ ⊥ ⊥ ⊥ + ⊥ ) , * , ⊥ ∀ ∈ ' ⊥ # # ⊥ ' + ' + }