Functional Analysis

About

I have always found the foundations of functional analysis, particularly the concept of the completion of a metric space, both fascinating and challenging. While I could grasp the ideas intuitively, I struggled to fully understand the formalism behind them. The main reason was that constructing the completion of a metric space depends on a prior concept: the completeness of the real numbers. In other words, I needed to revisit real analysis carefully before I could confidently approach functional analysis.

Another obstacle I faced was the variety of ways the real numbers can be constructed: through dedekind cut, Cauchy sequences, or Eudoxus constructions. Many textbooks mention these approaches without clarifying which one they adopt, which can be confusing. Eventually, I realized that constructing the real numbers explicitly is not always necessary, one can instead learn the axiom of real number. This axiomatic approach is more intuitive and concise, while still allowing us to rigorously prove key properties such as Cauchy completeness.

These experiences motivated me to write latex note to share a structured, accessible way of understanding these topics for those who might be facing similar challenges. There is one more point I wish I had understood earlier: mathematics is not built from formal proofs alone. The greatest mathematicians often began with geometric and visual intuition before formulating rigorous theorems. I used to focus solely on formal proofs, avoiding intuition for fear of being “less rigorous.” Over time, I realized that intuition is not the opposite of rigor, it is its foundation. Therefore, in these notes, I combine formal reasoning with geometric explanations, and I encourage you to embrace both perspectives while learning.

Contact

📧 Email: hngjesse@gmail.com
🔗 GitHub: hngjesse