## 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

- Conjugate Indices 1/p + 1/q = 1
- Young's Inequality
- Hölder's Inequality for ℓ1 and ℓp
- Hölder's Inequality for L1(Q) and Lp(E)
- Hölder's Inequality (General)
- Minkowski's Inequality for ℓ1, ℓp, and ℓ∞
- Minkowski's Inequality for L1(Q), Lp(E), and L∞(E)
- Minkowski's Inequality (General)
- Nested Property of the Lp(E) Spaces

## 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)

###### 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
- [[[Finite Dimensional Subspaces of Normed Linear Spaces

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

## 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

## 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

###### 4.4. Separable Spaces

- Separable Topological Spaces
- If X* is Separable then X is Separable
- If X is Reflexive then X is Separable IFF X* is Separable

###### 4.5. Topological Vector Spaces

## 5. Hilbert Spaces

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

- Inner Products and Inner Product Spaces
- 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
- Orthogonal Sets in an Inner Product Space
- Inner Products and Inner Product Spaces Review

###### 5.2. Hilbert Spaces

- Hilbert Spaces
- The Best Approximation Theorem for Hilbert Spaces
- 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
- Hilbert Spaces Review

###### 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
- Convergence Criterion for Series in Hilbert Spaces
- Hilbert Bases (Orthonormal Bases) for Hilbert Spaces
- Parseval's Identity for Inner Product Spaces
- Separability Criterion for Hilbert Spaces
- The Riesz Representation Theorem for Hilbert Spaces
- Orthogonal and Orthonormal Sets Review

# Queue

###### 2.2. Completeness of L^p(E

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

###### 2.1. Linear Operators on Linear Spaces

###### 2.2. Banach Spaces

###### 2.3. Finite-Dimensional Linear Spaces

- Equivalence of Norms in a Finite-Dimensional Linear Space X
- Every Linear Operator on a Finite-Dimensional Normed Linear Space is Bounded
- Isometries on Normed Linear Spaces X
- Isomorphism Linear Operators on Normed Linear Spaces X
- Two Finite-Dimensional Normed Linear Spaces of the Same Dimension are Isomorphic X
- Every Finite-Dimensional Normed Linear Space is a Banach Space X
- Riesz's Lemma X
- A Normed Linear Space is Finite-Dimensional If and Only If The Closed Unit Ball is Compact X
- Finite-Dimensional Linear Spaces Review X

###### 2.4. The Baire Category Theorem

- The Cantor Intersection Theorem for Complete Metric Spaces X
- The Baire Category Theorem for Complete Metric Spaces X
- Corollary to the Baire Category Theorem for Complete Metric Spaces X
- The Baire Category Theorem Review X

###### 2.5. The Open Mapping and Closed Graph Theorems

- Criterion for the Range of a BLO to be Closed when X is a Banach Space
- Closed Ranges of BLOs when Y is a Banach Space
- IFF Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces
- Second IFF Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces
- Open and Closed Mappings
- The Open Mapping Theorem
- Isomorphisms Between Banach Spaces X
- Equivalence of Norms on Banach Spaces X
- The Closed Graph Theorem
- The Open Mapping and Closed Graph Theorems Review

###### 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

###### 2.7. The Uniform Boundedness Principle

## 3. Linear Functionals, Duality

###### 3.1. Linear Functionals

- Linear Functionals on Linear Spaces
- The Algebraic Dual of a Linear Space
- Extensions of Linear Functionals on Subspaces of a Linear Space
- Expressing a Linear Functional as a Linear Combination of Other Linear Functionals
- The Topological Dual of a Normed Linear Space
- A Normed Linear Space is Finite-Dimensional IFF the Algebraic Dual and Topological Dual are the Same
- Linear Functionals Review

###### 3.2. Weak Topologies on Linear Spaces

- Weaker and Stronger Topologies X
- The F-Weak Topology on a Normed Linear Space X
- The W-Weak Topology on a Normed Linear Space X
- The Weak Topology (X*-Weak) Topology on a Normed Linear Space X

- The Second Topological Dual of a Normed Linear Space
- The Canonical Embedding of X into X**
- The Weak* (J(X)-Weak) Topology on the Topological Dual of a Normed Linear Space
- Reflexive Normed Linear Spaces X

###### 3.3. The Hahn-Banach Theorem

- Positively Homogeneous and Subadditive Functions X
- The Hahn-Banach Lemma X
- The Hahn-Banach Theorem (Real Version) X
- The Hahn-Banach Theorem (Complex Version) X

- Extensions of Linear Functionals with Equal Norms
- Finite-Dimensional Subspaces of Normed Linear Spaces have Topological Complements
- Criterion for a Point to be in the Closure of a Subspaces of Normed Linear Spaces
- The Canonical Embedding J is an Isometry

###### 3.4. Separable Spaces and Alaoglu's Theorem

- Helly's Theorem
- If a Normed Linear Space X* is Separable then X is Separable X
- A Reflexive Linear Space X is Separable IFF X* is Separable X
- Closed Subspaces of Reflexive Spaces are Reflexive
- Separable Criterion for the Compactness and Sequential Compactness of the Closed Unit Ball of X* in the Weak* Topology
- Alaoglu's Theorem
- Every Normed Linear Space is Isometrically Isomorphic to C(K) where K is a Compact Hausdorff Space
- Every Bounded Sequence in a Reflexive Space has a Weak* Convergent Subsequence
- Separable Spaces and Alaoglu's Theorem Review