Technology and Art

This is a quick note on proving the **Triangle Inequality** criterion of the following claim:

This post lists solutions to the exercises in the **Linear Operators section 2.6** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Compactness and Finite Dimension section 2.5** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Finite Dimensional Normed Spaces and Subspaces section 2.4** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Further Properties of Normed Spaces section 2.3** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Normed Space, Banach Space section 2.2** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Vector Space section 2.1** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Completion of Metric Spaces section 1.6** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Completeness Proofs section 1.5** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists solutions to the exercises in the **Convergence, Cauchy Sequences, and Completeness section 1.4** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This post lists assorted proofs from **Analysis**, without any particular theme.

This post lists solutions to many of the exercises in the **Open Set, Closed Set, Neighbourhood section 1.3** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

This article represents a (very short) collection of my **ongoing notes on proof tactics** I’ve found useful when I’ve been stuck trying to solve proof exercises. I aim to continue documenting these in as much detail as possible. These are mostly aids while building intuition about how to prove something, and gradually should become part of one’s mental lexicon.

This post lists solutions to many of the exercises in the **Distance Metrics section 1.2** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is a work in progress, and proofs may be refined over time.

Continuing my self-study of **Functional Analysis**, this post describes proofs for the following important inequalities in the subject:

- Young’s Inequality
- Hölder’s Inequality
- Minkowski’s Inequality

This post lists solutions to many of the exercises in the **Distance Metrics section 1.1** of *Erwin Kreyszig’s* **Introductory Functional Analysis with Applications**. This is definitely a work in progress, and proofs may be refined or added over time.

This article expands the groundwork laid in Kernel Functions: Functional Analysis and Linear Algebra Preliminaries to discuss some more properties and proofs for some of the properties of functions that we will use in future discussions on **Kernel Methods** in Machine Learning, including (but not restricted to) the construction of **Reproducing Kernel Hilbert Spaces**.

These are personal study notes, brief or expanded, complete or incomplete. Some concepts here will be alluded to in full-fledged Machine Learning posts.

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Continuing on my journey through **Real Analysis**, we will focus here on common **proof patterns** which apply to **irrational square roots**.
These patterns apply to the following sort of proof exercises:

Since I’m currently self-studying **Real Analysis**, I’ll be listing down proofs I either initially had trouble understanding, or enjoyed proving, here. These are very mathematical posts, and are for personal documentation, mostly.