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banach-spaces

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Banach Spaces

When to Use

Use this skill when working on banach-spaces problems in functional analysis.

Decision Tree

  1. Verify Banach space

    • Complete normed vector space
    • Check: every Cauchy sequence converges
    • z3_solve.py prove "completeness"

  2. Hahn-Banach Theorem

    • Extend bounded linear functionals
    • Separate convex sets
    • z3_solve.py prove "extension_exists"

  3. Open Mapping Theorem

    • Surjective bounded operator between Banach spaces is open
    • Consequence: bounded inverse exists
    • z3_solve.py prove "open_mapping"

  4. Closed Graph Theorem

    • T: X -> Y has closed graph implies T bounded
    • Strategy: verify graph closure, conclude boundedness
    • z3_solve.py prove "closed_graph_implies_bounded"

  5. Uniform Boundedness Principle

    • Pointwise bounded family of operators is uniformly bounded
    • Application: prove operator families are bounded

Tool Commands

Z3_Completeness

uv run python -m runtime.harness scripts/z3_solve.py prove "cauchy_sequence implies convergent"

Z3_Open_Mapping

uv run python -m runtime.harness scripts/z3_solve.py prove "T_surjective_bounded implies T_open"

Z3_Closed_Graph

uv run python -m runtime.harness scripts/z3_solve.py prove "graph_closed implies T_bounded"

Sympy_Norm

uv run python -m runtime.harness scripts/sympy_compute.py simplify "norm(alpha*x + beta*y)"

Key Techniques

From indexed textbooks:

  • [Introductory Functional Analysis with Applications] If (X, d) is a pseudometric space, we call a set B(xo; r) = {x E X I d(x, xo) < r} an open ball in X with center Xo and radius r. Note that this is analogous to 1. What are open balls of radius 1 in Prob.
  • [Measure, Integration Real Analysis (... (Z-Library)] Section 5C Lebesgue Integration on Rn 11 Suppose E is a subset of Rm Rn and Rm : (x, y) x E for some y . Dene f : R2 R by = (0, 0), (a) Prove that D1(D2 f ) and D2(D1 f ) exist everywhere on R2. Show that D1(D2 f ) (c) Explain why (b) does not violate 5.
  • [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] The Hahn-Banach Theorem has a rather humble nature. The only mathematical con- cepts needed for its statement are linear spaces and linear, subadditive, and positively homogeneous functionals. Besides Zorn’s Lemma, its proof relies on nothing more than the rudimentary properties of the real numbers.
  • [Introductory Functional Analysis with Applications] If in a normed space X, absolute convergence of any series always implies convergence of that series, show that X is complete. Show that in a Banach space, an absolutely convergent series is convergent. Schauder basis) Show that if a normed space has a Schauder basis, it is separable.
  • [Introductory Functional Analysis with Applications] What are the adjoints of a zero operator 0 and an identity operator I? Annihllator) Let X and Y be normed spaces, T: X - Y a bounded linear operator and -M = (¥t( T), the closure of the range of T. Fundamental Theorems for Normed and Banach Spaces To complete this discussion, we should also list some of the main differences between the adjoint operator T X of T: X ~ Y and the Hilbert-adjoint operator T* of T: Hi ~ H 2 , where X, Yare normed spaces and Hi> H2 are Hilbert spaces.

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.

Source

git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/functional-analysis/banach-spaces/SKILL.mdView on GitHub

Overview

This skill provides a practical workflow for tackling Banach-space problems in functional analysis. It centers on verifying completeness, applying core theorems (Hahn-Banach, Open Mapping, Closed Graph, Uniform Boundedness), and using solver-based proofs to justify boundedness and extensions.

How This Skill Works

Begin by confirming the space is Banach (complete). Use a structured decision tree to select the appropriate theorem, then invoke the corresponding solver (e.g., for completeness, Hahn-Banach extension, Open Mapping, or Closed Graph) to generate a proof sketch or verification. The process yields concrete boundedness, openness, or extension results that drive the final argument.

When to Use It

  • You need to confirm a normed space is Banach by checking completeness (every Cauchy sequence converges).
  • You must extend a bounded linear functional or separate convex sets using Hahn-Banach.
  • You need to deduce openness and possibly a bounded inverse for a surjective bounded operator via Open Mapping.
  • You want to infer operator boundedness from a closed graph through the Closed Graph Theorem.
  • You must show a family of operators is uniformly bounded using the Uniform Boundedness Principle.

Quick Start

  1. Step 1: Identify the problem class and select the relevant theorem (completeness, Hahn-Banach, Open Mapping, Closed Graph, or Uniform Boundedness).
  2. Step 2: Run the corresponding solver/proofs to validate the property (e.g., cauchy_sequence implies convergent for completeness; extension_exists for Hahn-Banach; open_mapping for Open Mapping; graph_closed implies T_bounded for Closed Graph).
  3. Step 3: Interpret the solver output to assert boundedness, openness, or extendibility, and document the resulting bounds and implications.

Best Practices

  • Verify completeness early to ensure the space is Banach before applying theorems.
  • When using Hahn-Banach, ensure the functional is bounded and the extension is well-defined on the whole space.
  • Apply Open Mapping only after confirming surjectivity to conclude openness and the existence of a bounded inverse if applicable.
  • Use the Closed Graph Theorem by first checking that the graph of the operator is closed.
  • Leverage Uniform Boundedness to move from pointwise boundedness to uniform boundedness for operator families.

Example Use Cases

  • Proving that L^p spaces (1 ≤ p ≤ ∞) are Banach to justify applying extension and mapping theorems.
  • Extending a bounded linear functional from a subspace to the entire space via Hahn-Banach.
  • Using the Open Mapping Theorem to deduce that a surjective bounded operator between Banach spaces has a bounded inverse.
  • Applying the Closed Graph Theorem to conclude that a linear operator with a closed graph is automatically bounded.
  • Using Uniform Boundedness to show a family of operators is uniformly bounded when each operator is pointwise bounded on a Banach space.

Frequently Asked Questions

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