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Appendix A - Functional Analysis Cheat Sheet (DRAFT) by

NM4BIP Appendix A - Basic Concepts in Functional Analysis

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Normed spaces and Banach spaces

Strong conver­gence
||xn - x||X -> 0
Continuous embedding
X ⊂ Y and ||x||Y ≤ C||x||X
Banach space = complete normed space
Closure = add all limits of convergent sequences from the set

Bounded linear operators

D(A) ⊂ X, N(A) ⊂ X, R(A) ⊂ Y
A ∈ L(X,Y) <=> A linear and bounded
Bounde­dness
||Ax||Y ≤ C||x||X
Operator norm
||A|| = supx≠1 ||Ax||Y／||x||X
Inverse bounded ⟺ c||x||X ≤ ||Ax||Y (c > 0)

Conver­gence, Banach­-St­einhaus

Pointwise
∀x, Anx -> Ax ∈ Y
Uniform
||An - A||L(X,Y) -> 0
Banach-
Steinhaus
pointwise bounded family => supi ||Ai|| < ∞

Hilbert spaces, weak conver­gence

Hilbert space = Banach space + scalar product
Weak conver­gence
xn⇀x ⟺ <xn,z> ->
<x,­z>,∀z ∈ X
Strong => weak. In infinite dims: weak ⇏ strong
Weak + conver­gence of norms => strong
 

Riesz repres­ent­ation, adjoint operators

Riesz
λ(x) = <zλ, x>
Adjoint
<A*y, x>X = <y, Ax>Y
Matrix case
A* = AT over ℝ
Properties
(A*)* = A, ||A*|| = ||A||, ||A*A|| = ||A||2

Orthog­onality and projec­tions

U⊥ = {x ∈ X : <x,­u> = 0 ∀u ∈ U}
U⊥ is always closed
If U closed:
X = U ⊕ U⊥
If U not closed:
(U⊥)⊥ = cl(U)
Projection
z = PUx
⟺ z ∈ U and x - z ∈ U⊥

Range/­kernel identities

R(A)⊥ = N(A*)
N(A*)⊥ = cl(R(A))
R(A*)⊥ = N(A)
N(A)⊥ = cl(R(A*))
Kernel closed, range may be non-closed

Orthon­ormal systems and bases

Orthon­ormal
<ui,uj> = δij
Bessel
∑j|<x,uj>|2 <= ||x||2
ONB expansion
x = ∑j<x,uj>uj
Projection
PUx = ∑j<x,uj>uj
 

Compact operators

Compact
bounded sequence xn -> Axn has a convergent subseq­uence
Equivalent
weak conver­gence in X -> strong conver­gence after A in Y
Finite­-di­men­sional range => compact
Id compact ⟺ dim(X) < ∞
A,S ∈ L(X,Y), A or S compact => A∘S compact
A compact ⟺ A* compact

Example: integral operators

Kx(t) = ∫01 k(s,t)­x(s)ds
Approx­imate k by piecew­ise­-co­nstant kn
Kn finite rank => Kn compact;
Kn -> K in operator norm => K compact.
Self-a­djoint ⟺ k(s,t) = k(t,s) a.e. (real case)

Spectral theorem (for compact, K=K*)

Kx = ∑nλn<x,un>un, with λn -> 0
Kun = λnun,
{un} ONB for cl(R(K))
||K|| = |λ1| when ordered by |λ1|≥|λ2|≥⋯≥0
Inverse danger: divide by small λn