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#### Theorem 1.1 Helly, Hahn-Banach analytic form

#### Result

Hahn-Banach Theorem admits an extension of the linear functional on a linear subspace to the whole space which is bounded by a maximal element admitted by the Zorn’s Lemma.

#### Proof(See Appendix Above)

Hahn-Banach Theorem admits an extension of the linear functional on a linear Subspace to the whole space which is bounded by a maximal element admitted by the Zorn’s Lemma.

#### Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

#### Related Topics

- Definitions
- Vector Space
- Positive Homogeneous
- Additive(subadditive)
- Linear Space
- Linear Functional

- Theorems/Corollary/Propositions/Lemmas
- Hahn-Banach, 1st geometric form
- Hahn-Banach, 2nd geometric form(FASSPDE)
- Frenchel – Moreau(FASSPDE)

### Corollary 1.1.1

#### Statement

#### Result

This corollary makes it possible for the norm in the whole space of the extension of g into the subspace where g is in, and the equality always hold.

#### Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

### Corollary 1.1.2

#### Statement

#### Remark

#### Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas

### Corollary 1.1.3

#### Statement

#### Remark

#### Reference

Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Chapter 1

#### Related Topics

- Definitions
- Theorems/Corollary/Propositions/Lemmas