# Functional Analysis Topics

## 1. Classical Linear Spaces

###### 1.1. Linear Spaces (Vector Spaces)

- Linear Spaces
- Normed Linear Spaces
- Subspaces of Linear Spaces
- Products of Linear Spaces
- Continuity of the Norm on Normed Linear Spaces
- Linearly Independent Sets and Spanning Sets in Linear Spaces
- Finite-Dimensional Linear Spaces

###### 1.2. Normed Linear Spaces

- The ℓ1 Sequences Normed Linear Space
- The ℓp Sequences Normed Linear Space
- The ℓ∞ Sequences Normed Linear Space
- The c0 Sequences Normed Linear Space
- ℓ1, ℓp, ℓ∞, and c0 Sequences Normed Linear Spaces Review

- The L1(E) Normed Linear Space
- The Lp(E) Normed Linear Space
- The L∞(E) Normed Linear Space
- L1(E), Lp(E), and L∞(E) Normed Linear Spaces Review

###### Young's, Hölder's, and Minkowski's Inequalities

## 2. Completeness, Linear Functionals, Linear Operators, and Compactness of the Closed Unit Ball

###### 2.1. Completeness

- Complete Metric Spaces
- Banach Spaces
- Absolutely Summable Series
- Absolute Summability Criterion for Completeness

- Lemma to the Riesz-Fischer Theorem (1 ≤ p < ∞)
- Lemma to the Riesz-Fischer Theorem (p = ∞)
- The Riesz-Fischer Theorem

###### 2.2. Linear Functionals, Bounded Linear Functionals, and Dual Spaces

- Linear Functionals and Bounded Linear Functionals
- The Operator Norm on the Set of Bounded Linear Functionals
- The Dual Space of a Normed Linear Space
- The Riesz Representation Theorem for Lp(E)
- Unbounded Linear Functionals

###### 2.3. Linear Operators, Bounded Linear Operators, and the Space of Bounded Linear Operators

- Linear Operators and Bounded Linear Operators ( Examples 1 )
- The Operator Norm on the Set of Bounded Linear Operators
- The Range and Kernel of Linear Operators
- The Space of Bounded Linear Operators
- Criterion for B(X, Y) to be a Banach Space
- Bounded Linear Operators from X to X

###### 2.4. Isomorphisms and Isometries

- Isometries Between Normed Linear Spaces
- Isomorphisms Between Normed Linear Spaces
- Criterion for a Linear Map to be an Isomorphism Between Normed Linear Spaces
- Norm Equivalence
- Equivalence of Norms in a Finite-Dimensional Linear Space
- Corollaries to the Equivalence of Norms in a Finite-Dimensional Linear Space
- Boundedness of Linear Operators on Finite-Dimensional Normed Linear Spaces
- Isomorphic Finite-Dimensional Normed Linear Spaces
- Finite Dimensional Normed Linear Spaces are Banach Spaces

###### 2.5. Compactness of the Closed Unit Ball

- Riesz's Lemma
- Closed Unit Ball Criterion for Finite Dimensional Normed Linear Spaces
- Examples of Closed Unit Balls that are NOT Compact

###### 2.6. Algebraic and Topological Complements of Linear Subspaces

- Projection/Idempotent Linear Operators
- Algebraic Complements of Linear Subspaces
- Topological Complements of Normed Linear Subspaces
- Topological Complement Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces
- T(X) Finite Co-Dimensional Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces
- Algebraic and Topological Complements of Linear Subspaces Review

## 3. The Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle, and Hahn-Banach Theorem

###### 3.1. The Baire Category Theorem

- Nowhere Dense Subsets of Metric Spaces
- Metric Spaces of the First Category and Second Category
- The Cantor Intersection Theorem for Complete Metric Spaces
- The Baire Category Theorem for Complete Metric Spaces
- Corollary to the Baire Category Theorem for Complete Metric Spaces
- The Baire Category Theorem Review

###### 3.2. The Open Mapping Theorem

- Open Maps and Closed Maps Between Metric Spaces
- The Open Mapping Theorem
- Corollaries to the Open Mapping Theorem

###### 3.3. The Closed Graph Theorem

- The Closed Graph Theorem
- An Example of a Linear Operator with a Closed Graph that is Unbounded
- Corollaries to the Closed Graph Theorem

###### 3.4. The Uniform Boundedness Principle

- The Lemma to the Uniform Boundedness Principle
- The Uniform Boundedness Principle
- Corollaries to the Uniform Boundedness Principle

###### 3.5. The Hahn-Banach Theorem

- Sublinear Functionals
- The Hahn-Banach Lemma
- The Hahn-Banach Theorem (Real Version)
- The Hahn-Banach Theorem (Complex Version)
- Corollaries to the Hahn-Banach Theorem

###### 3.5. Adjoint Linear Operators Between Banach Spaces

## 4. The Weak Topology and Weak* Topology

###### 4.1. Topological Spaces

- The Fundamentals of Topological Spaces
- Examples of Topological Spaces
- The Subspace and Product Topologies
- Bases for a Topology

###### 4.2. The Weak and Weak-* Topologies

- Weaker and Stronger Topologies
- The Weak Topology Induced by F
- Expressing a Linear Functional as a Linear Combination of Other Linear Functionals
- The Weak Topology Induced by W ⊆ X♯
- The Weak Topologies on X and X*
- The Weak-* Topology on X*
- A Comparison of the Weak and Weak* Topologies
- Coincidence of the Weak and Norm Topologies on a Normed Linear Space
- Every Weakly Convergent Sequence in X is Norm Bounded
- Every Weakly Compact Set in X is Norm Bounded

###### 4.3. Reflexive Spaces

- The Natural Embedding J
- The Natural Embedding J is an Open Map
- Reflexive Normed Linear Spaces
- Closed Subspaces of Reflexive Spaces are Reflexive
- IFF Criterion for Reflexivity of a Normed Linear Space

###### 4.4. Helley's Theorem

- Separable Topological Spaces
- If X* is Separable then X is Separable
- If X is Reflexive then X is Separable IFF X* is Separable
- Helley's Theorem
- Every Bounded Sequence in a Reflexive Space X has a Weakly Convergent Subsequence
- Separable Criterion for the Weak-* Compactness and Weak-* Sequential Compactness of the Closed Unit Ball of X*

###### 4.5. Topological Vector Spaces

- Topological Vector Spaces (TVS)
- Convex Subsets of Vector Spaces
- Locally Convex Topological Vector Spaces (LCTVS)
- X with the W-Weak Topology is a LCTVS
- The Closure of a Convex Set is Closed in a LCTVS
- Continuity of Linear Functions on LCTVS

###### 4.6. Separation

- The Separation of Two Sets by a Linear Functional
- The Separation Theorems
- Separation of a Subspace by a Continuous Linear Function on LCTVS
- Mazur's Theorem

###### 4.7. The Krein-Milman Theorem

- Extreme Subsets and Extreme Points of a Set in a LCTVS
- Extreme Points of the Closed Unit Ball of a Normed Linear Space
- The Closed Convex Hull of a Set in a LCTVS
- The Krein-Milman Lemma
- The Krein-Milman Theorem

###### 4.8. Alaoglu's Theorem, Kakutani's Theorem, Goldstine's Theorem, and the Eberlein-Smulian Theorem

## 5. Inner Product Spaces

###### 5.1. Inner Products and Inner Product Spaces

- Inner Products and Inner Product Spaces
- The Inner Product on Rn and Cn
- The Inner Product on ℓ2 and L2(E)
- The Cauchy-Schwarz Inequality for Inner Product Spaces
- The Normed Space Induced by an Inner Product
- The Parallelogram Identity for the Norm Induced by an Inner Product
- Proof that ℓp when p ≠ 2 is an Inner Product Space
- Orthogonal Vectors in an Inner Product Space
- The Orthogonal Subspace S⟂ of an Inner Product Space

###### 5.2. Hilbert Spaces

- Hilbert Spaces
- The Best Approximation Theorem for Hilbert Spaces
- Proof that ℓp when p ≠ 2 is NOT a Hilbert Space
- The Riesz Representation Theorem for Hilbert Spaces
- The Radon-Riesz Theorem
- Weak Convergence in Hilbert Spaces
- Every Bounded Sequence in a Hilbert Space has a Weakly Convergent Subsequence
- Algebraic Complements of Closed Subspaces of Hilbert Spaces
- Density of the Span of Closed Subsets in Hilbert Spaces
- The Orthogonal Projection of a Hilbert Space onto a Closed Subspace

###### 5.3. Orthogonal and Orthonormal Sets

- Orthogonal and Orthonormal Sets in Inner Product Spaces
- The Pythagorean Identity for Inner Product Spaces
- Bessel's Inequality for Inner Product Spaces
- Corollaries to Bessel's Inequality
- Weak Convergence of Orthonormal Sequences to 0 for Inner Product Spaces
- Convergence Criterion for Series in Hilbert Spaces

###### 5.4. Orthonormal Bases

## 6. Normed Algebras

###### 6.1. Algebras, Normed Algebras, and Banach Algebras

- Algebras over F
- Normed Algebras and Seminormed Algebras
- Continuity of Addition, Scalar Multiplication, and Multiplication on Normed Algebras
- The Completion of a Normed Algebra
- The Reversed Algebra of an Algebra
- The Operator Algebra of Bounded Linear Operators on X
- The Operator Algebra of Compact Bounded Linear Operators on X
- The Normed Algebra ℓ∞(E, A) for a Nonempty Set E and a Normed Algebra A
- The Group Algebra of R, L1(R)
- The Discrete Group Algebra of a Group G, ℓ1(G)
- The Discrete Semigroup Algebra of a Semigroup S, ℓ1(S, α)
- List of Algebras

###### 6.2. Invertible and Singular Elements in an Algebra

- Invertible and Singular Elements in an Algebra
- For Normed Algebras with Unit, the Unit Can be Assumed to Have Norm 1
- The Commutant (Centralizer) of a Set in an Algebra
- The Group of Invertible Elements in an Algebra
- Basic Theorems Regarding The Group of Invertible Elements in an Algebra
- The Spectral Radius of a Point in a Normed Algebra
- Invertibility of 1 - a When r(a) < 1 in a Banach Algebra with Unit
- Inv(A) is an Open Subset of A in a Banach Algebra with Unit
- Topological Divisors of Zero in a Normed Algebra
- Boundary Points of Inv(A) are Topological Divisors of Zero in a Banach Algebra with Unit
- Permanently Singular Elements in a Banach Algebra with Unit

###### 6.3. Quasi-Invertible and Quasi-Singular Elements in an Algebra

- The Unitization of a Normed Algebra
- Quasi-Invertible and Quasi-Singular Elements in an Algebra
- Algebras with Unit - x Has Quasi-Inverse y IFF 1 - x Has Inverse 1 - y
- Algebras - x Has Quasi-Inverse y IFF (0, 1) - (x, 0) Has Inverse (0, 1) - (y, 0)
- Algebras - xy is Quasi-Invertible IFF yx is Quasi-Invertible
- Quasi-Invertibility of x When r(x) < 1 in a Banach Algebra
- q-Inv(A) is an Open Subset of A in a Banach Algebra
- Boundary Points of q-Inv(X) are Quasi-Singular in a Banach Algebra

###### 6.4. The Spectrum of an Element in an Algebra

- Homomorphisms, Monomorphisms, and Isomorphisms Between Algebras
- The Spectrum of an Element in an Algebra over C
- Basic Theorems Regarding the Spectrum of an Element in an Algebra over C
- The Spectrum of an Element Applied to a Polynomial in an Algebra over C
- The Spectrum of an Element in a Normed or Banach Algebra over C
- r(xy) = r(yx) When A is a Banach Algebra over C
- Examples of Spectrums of Elements in an Algebra

###### 6.5. Ideals

- Ideals in a Linear Space
- Basic Theorems Regarding Ideals in an Algebra 1
- Basic Theorems Regarding Ideals in an Algebra 2

- The Difference Space of a Linear Space X Modulo a Linear Subspace L
- The Quotient Algebra of an Algebra X Modulo a Two-Sided Ideal J

###### 6.6. Left A-Modules, Right A-Modules, and A-Bimodules

- Left A-Modules, Right A-Modules, and A-Bimodules
- Examples of Modules - Left Ideals J are Left A-Modules, Right Ideals J are Right A-Modules
- Examples of Modules - For Closed Left Ideals J, A/J is a Normed Left A-Module
- Examples of Modules - For Normed Left A-Modules M, M is a Normed Left (A + F)-Module
- Examples of Modules - The Dual Banach Right Module of a Normed Left A-Module M
- Examples of Modules - For Normed Algebras A, A* is a Banach Left A**-Module
- Examples of Modules - For Banach Left A-Modules M, c0(M) is a Banach Left A-Module
- Examples of Modules - For Banach Left A-Modules M, l1(M) is a Banach Left A-Module

###### 6.7. Approximate Identities

- Nets in a Normed Algebra
- Approximate Identities in a Normed Algebra
- Weak Approximate Identities in a Normed Algebra
- The Quasi-Product of Bounded Nets with Left/Right Approximate Identities
- Approximate Identities in A for X when X is Banach Left A-Module 1
- Approximate Identities in A for X when X is Banach Left A-Module 2
- Approximate Identities in A for X when X is Banach Left A-Module 3
- Approximate Identities in A for X when X is Banach Left A-Module 4
- The First and Second Arens Products on A**
- A** (with First Arens Product) Has a Right Identity IFF A Has a Bounded Right Approximate Identity

###### 6.8. Commutative Subsets of an Algebra

- Commutative Subsets of an Algebra
- Maximal Commutative Subsets of an Algebra
- The Spectrum of an Element in a Maximal Commutative Subset of an Algebra
- Conditions for a Complex Banach Algebra with Unit to be Commutative

###### 6.9. Multiplicative Linear Functionals on a Banach Algebra

- Multiplicative Linear Functionals on a Banach Algebra
- For Algebras with Unit A - ker(f) ⊆ Sing(A) for all Multiplicative Linear Functionals f
- For Algebras A - ker(f) is a Two-Sided Ideal of A for all Multiplicative Linear Functionals f
- The Gelfand-Mazur Theorem
- Characterization of Maximal Modular Two-Sided Ideals of Codimension 1 in a Banach Algebra over C
- Characterization of Maximal Modular Two-Sided Ideals in a Commutative Banach Algebra over C
- Jordan Functionals on a Banach Algebra
- Equivalent Criteria for a Linear Functional to be Multiplicative in a Banach Algebra
- The Spectrum of an Element in a Commutative Banach Algebra over C

###### 6.10 The Gelfand Representation of a Commutative Banach Algebra

- The Carrier Space of a Commutative Banach Algebra
- The Gelfand Representation of a Commutative Banach Algebra
- The Radical of a Commutative Banach Algebra
- Semi-Simple Commutative Banach Algebras
- Continuity of Homomorphisms from a Commutative Banach Algebra to a Semi-Simple Commutative Banach Algebra

###### 6.11. Tensor Products

- Bilinear Mappings and Multilinear Mappings
- Examples of Bilinear Mappings
- The Algebraic Tensor Product of Two Normed Linear Spaces
- Properties of Elementary Tensors of Normed Linear Spaces
- Linear Independence Properties of Tensor Products of Normed Linear Spaces
- X⊗Y and Y⊗X are Isomorphic Linear Spaces
- Criteria for a Tensor u in X⊗Y to be 0
- Tensor Products with Direct Sums of Normed Linear Spaces
- The Existence of a Linear Map σ on X⊗Y to Z that Matches a Bilinear Map on X×Y to Z
- The Trace Linear Functional on X*⊗X
- The Weak Tensor Product of X⊗Y
- The Projective Tensor Product of X⊗Y
- Alternative Definitions of the Projective Tensor Norm on X⊗Y
- (X⊗pY)* and BL(X, Y; F) are Isometrically Isomorphic
- The Closed Unit Ball of X⊗pY
- ℓ1⊗pX is Isometrically Isomorphic to ℓ1(X) for Banach Spaces X
- Tensor Products of Linear Operators of Normed Linear Spaces
- Cross Norms on X⊗Y
- Embedding Subspaces A of BL(X) and B of BL(Y) Into BL(X⊗pY)
- The Algebraic Tensor Product of Two Normed Algebras
- For Commutative Banach Algebras A and B over C - ΦA×ΦB is Homeomorphic to ΦA⊗pB
- For Semi-Simple Commutative Banach Algebras A and B - A⊗B is Semi-Simple

###### 6.12. Amenability of Banach Algebras over C with Unit

###### Queue

- Inner Products and Inner Product Spaces Review
- Hilbert Spaces Review
- Orthogonal and Orthonormal Sets Review

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###### References

- 1. Real Analysis (3rd Edition) by Halsey Royden.

- 2. Real Analysis (4th Edition) by Halsey Royden and Patrick Fitzpatrick.

- 3. Complete Normed Algebras by Frank F. Bonsall and John Duncan.

- 4. Introduction to Tensor Products of Banach Spaces by Raymond A. Ryan.