Open Mapping and Closed Graph

Key idea

Pillars of functional analysis.

In depth

A bounded linear surjection between Banach spaces is open, so a bijective bounded operator has a bounded inverse. The closed graph theorem says a linear map with closed graph is bounded. Both rely on completeness and Baire category.

Worked example

If a bijection T between Banach spaces is bounded, then T⁻¹ is automatically bounded.

Real-world uses

• Proving that solutions of linear equations depend stably on the data in engineering

Common mistake

Applying them to non-complete spaces.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.