CNAM UE MVA 210 Ph. Durand Algèbre et analyse tensorielle

E K
E N E R+
∀x∈E×E N(x) = 0 ⇒x= 0
∀(λ, x)∈K×E N(λx) = |λ|N(x)
∀(x, y)∈E2N(x+y)≤N(x) + N(y)
Rn(x1, x2, ..;xn)
kxkp= (|x1|p+...+|xn|p)1
pkxk∞=Sup1≤i≤n|xi|
E=C([a, b],R)
[a, b]
kfk2= (Rb
a|f(x)|dx)1
2
kfk∞=Supx∈[a,b]|f(x)|
C([a, b],R)

kfk2= (Rb
a|f(x)|dx)1
2
(E, N)d(x, y) = N(x−y)d
∃α, β ∀x∈E αN1(x)< N2(x)< βN1(x).
Rn
p= 2
ϕ E

ϕ(x, y) = ϕ(y, x)
ϕ(x+y, z) = ϕ(x, z) + ϕ(y, z)
ϕ(λx, y) = λϕ(x, y)
ϕ(x, x)≥0
ϕ(x, x) = 0 ⇒x= 0
Rnϕ(x, y) = x1y1+...xnynx= (x1, ...xn)y= (y1, ...yn)
C([a, b],R)ϕ(x, y) = Rb
af(x)g(x)dx)
< x, y >=c2.tt0−x1y1−x2y2−x3y3
ϕ(x, x)
(X1,O1) (X2,O2)fX1
X2f l x a
∀V∈ V(l)∃U∈ V(a)f(U)⊂V
f a l =f(a)f
a
X1=X2=R
limx→af(x) = l

l]l−ε, l +ε[
a]a−α, a +α[
∀ > 0∃α > 0f(]a−α, a +α[) ⊂]l−ε, l +ε[
∀ > 0∃α > 0|x−a|< α ⇒ |f(x)−l|< ε
limn→+∞f(n) = l
Vn={n+ 1, n + 2, ...}
l]l−ε, l +ε[
+∞VN={N+ 1, N + 2, ...}
∀ > 0∃N > 0f({N+ 1, N + 2, ...})⊂]l−ε, l +ε[
∀ > 0∃N > 0n>N ⇒ |f(n)−l|< ε
un=f(n)
RnRp
N1N2
f l x x0
∀ > 0∃α > 0N1(x−x0)< α ⇒N2(f(x)−l)< ε
x0limx→x0f(x) = f(x0)