Teoremi e definizioni di Advanced Mathematics

THEOREMS AND DEFINITIONS

CHAPTER 7

DEF. 245) Given a vector space V, a positive function ||·|| is said to be a norm if:

• i) ||v|| = 0 ⇒ v = 0
• ii) ||v + w|| ≤ ||v|| + ||w|| ∀v, w ∈ V
• iii) ||a⋅v|| = |a| ||v|| ∀a ∈ ℝ, ∀v ∈ V

DEF. 247) A vector space endowed with a norm ||·|| is called normed vector space.

LM. 248) Let (V, ||·||) be a normed vector space. The function d: V x V → ℝ defined by

d(v,w) = ||v - w|| ∀v, w ∈ V

is a distance, with two additional properties:

• i) Translation invariance (d(v+x, w+x) = d(v,w))
• ii) Homogeneity (d(αv, βw) = |αβ| d(v,w))

Not all distances in metric spaces are induced by norm

DEF. 252) A normed vector space whose metric is complete is called a Banach space.

LM. 257) We have |||v|| - ||w||| ≤ ||v - w||

PROP. 259) A linear operator T: V₁ → V₂ between N.V.S. is continuous at a point v ∈ V₁ ⇔ it is continuous on V₁.

DEF. 260) An operator T: V₁ → V₂ is called bounded if ∃K > 0 s.t. ||T(v)||₂ ≤ K||v||₁ ∀v ∈ V₁

LM. 261) A linear operator T: V₁ → V₂ between N.V.S. is bounded ⇒ the image of any bounded subset of V₁ is a bounded subset of V₂.

TH. 263) A linear operator T: V₁ → V₂ between N.V.S. is continuous ⇔ it is bounded,

DEF) The topological dual V* of a N.V.S. is the set

THEOREMS AND DEFINITIONS

CHAPTER 7

DEF. 245) Given a vector space V, a positive function ||.|| is said to be a norm if:

• i) ||v|| = 0 ⇒ v = 0
• ii) ||v+w|| ≤ ||v|| + ||w|| ∀v,w ∈ V
• iii) ||αv|| = |α| ||v|| ∀α ∈ ℝ, ∀v ∈ V

DEF. 247) A vector space endowed with a norm ||.|| is called a normed vector space.

LM. 248) Let (V, ||.||) be a normed vector space. The function d: V×V → ℝ defined by

• d(v,w) = ||v-w|| ∀v,w ∈ V

is a distance, with two additional properties:

• i) Translation invariance (d(v+x, w+x) = d(v,w))
• ii) Homogeneity (d(αv, αw) = |α| d(v,w))

Not all distances in metric spaces are induced by norms.

DEF. 252) A normed vector space whose metric is complete is called a Banach space.

LM. 257) We have ||v|| - ||w|| ≤ ||v-w||

PROP. 259) A linear operator T: V₁ → V₂ between N.V.S. is continuous at a point y ∈ V₄ ⇔ it is continuous on V₁.

DEF. 260) An operator T: V₄ → V₂ between N.V.S. is called bounded if ∃K>0 s.t.

• ||T(v)||₂ ≤ K ||v||₄ ∀v ∈ V₄

LM. 261) A linear operator T: V₄ → V₂ between N.V.S. is bounded ⇒ the image of any bounded subset of V₄ is a bounded subset of V₂.

TH. 263) A linear operator T: V₄ → V₂ between N.V.S. is continuous ⇒ it is bounded.

DEF.) The topological dual V* of a N.V.S. is the set

OF ALL CONTINUOUS AND LINEAR FUNCTIONALS DEFINED ON V

IT IS A VECTOR SUBSPACE OF THE ALGEBRAIC DUAL V'. THE

DEFINITION IS ANALOGOUS FOR THE SET OF ALL CONTINUOUS

AND LINEAR OPERATORS BETWEEN N.V.S. V₁ AND V₂, B(V₁,V₂).

LM. 267) LET T∈B(V₁, V₂). THE SET

{k>0 | ‖T(v)‖₂ ≤ k ‖v‖₁ ; ∀ v ∈ V₁}

HAS A MINIMUM.

PROP. 269) THE FUNCTIONAL ‖ · ‖ : B(V₁, V₂) →ℝ, DEFINED BY

‖T‖ = min {k ≥ 0 | ‖T(v)‖₂ ≤ k‖v‖₁, ∀ v ∈ V₁}

IS A NORM.

PROP. 270) LET T ∈ B(V₁, V₂). IT HOLDS:

‖T‖ = sup{‖T(v)‖₂‖v‖₁} ∀v≠0

= sup{‖T(v)‖₂}∀v: ‖v‖₁ = 1

= sup{‖T(v)‖₂}∀v∈S₁

TH. 273) IF V₂ IS A BANACH SPACE, ALSO THE N.V.S. B(V₁, V₂) IS

A BANACH SPACE.

PROPERTIES)

‖T + S‖ ≤ ‖T‖ + ‖S‖

‖TS‖ ≤ ‖T‖ ‖S‖

(S ∈ B(V₁, V₂), T ∈ B(V₂, V₃))

★ CHAPTER 8

DEF.) IN ITS GENERAL FORM, AN EQUATION CAN BE WRIT

AS

ℒ(x) = y₀

DEF.) THE INVERSE CORRESPONDENCE ℒ^-1 : 2^X → Y IS DEFINED BY

ℒ^-1(y) = {x∈X | ℒ(x) = y} ∀y∈Y

DEF.) ℒ IS WEAKLY INVERTIBLE AT y∈Y IF ℒ^-1(y) IS NOT

EMPTY (I.E. ∃x SURJECTIVE W.R.T. y∈Y), ℒ IS INVERTIBLE

AT y∈Y IF ℒ^-1(y) IS A SINGLETON (I.E. ∀ y BIJECTIVE W.R.T. y∈Y). IF THESE HOLD ∀ y∈Y, ℒ IS (WEAKLY) INVERTIBLE.

{IT IS WELL POSED IF → IS A CONTINUOUS FUNCTION

DEF.)

γ = ϑ(x; m)

effect