mathematics

Open mapping theorem (functional analysis)

Open mapping theorem (functional analysis) is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Open mapping theorem (functional analysis) rather than just read about it. In short: In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomor…

Key takeaways

Open mapping theorem (functional analysis) belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Open mapping theorem (functional analysis) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Open mapping theorem (functional analysis) from memory before moving on to harder problems.

Reference excerpt

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator T {\displaystyle T} from one Banach space to another has bounded inverse T − 1 {\displaystyle T^{-1}} .

Statement and proof

The proof here uses the Baire category theorem, and completeness of both E {\displaystyle E} and F {\displaystyle F} is essential to the theorem. The statement of the theorem is no longer true if either space is assumed to be only a normed vector space; see § Counterexample. The proof is based on the following lemmas, which are also somewhat of independent interest. A linear map f : E → F {\displaystyle f:E\to F} between topological vector spaces is said to be nearly open if, for each neighborhood U {\displaystyle U} of zero, the closure f ( U ) ¯ {\displaystyle {\overline {f(U)}}} contains a neighborhood of zero. The next lemma may be thought of as a weak version of the open mapping theorem.

Proof: Shrinking U {\displaystyle U} , we can assume U {\displaystyle U} is an open ball centered at zero. We have f ( E ) = f ( ⋃ n ∈ N n U ) = ⋃ n ∈ N f ( n U ) {\displaystyle f(E)=f\left(\bigcup _{n\in \mathbb {N} }nU\right)=\bigcup _{n\in \mathbb {N} }f(nU)} . Thus, some f ( n U ) ¯ {\displaystyle {\overline {f(nU)}}} contains an interior point y {\displaystyle y} ; that is, for some radius r > 0 {\displaystyle r>0} ,

B ( y , r ) ⊂ f ( n U ) ¯ . {\displaystyle B(y,r)\subset {\overline {f(nU)}}.}

Then for any v {\displaystyle v} in F {\displaystyle F} with ‖ v ‖ < r {\displaystyle \|v\|<r} , by linearity, convexity and ( − 1 ) U ⊂ U {\displaystyle (-1)U\subset U} ,

v = v − y + y ∈ f ( − n U ) ¯ + f ( n U ) ¯ ⊂ f ( 2 n U ) ¯ {\displaystyle v=v-y+y\in {\overline {f(-nU)}}+{\overline {f(nU)}}\subset {\overline {f(2nU)}}} , which proves the lemma by dividing by 2 n {\displaystyle 2n} . ◻ {\displaystyle \square } (The same proof works if E , F {\displaystyle E,F} are pre-Fréchet spaces.) The completeness on the domain then allows to upgrade nearly open to open.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Open mapping theorem (functional analysis)

Start with the simplest possible case. Write down what Open mapping theorem (functional analysis) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Open mapping theorem (functional analysis) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Open mapping theorem (functional analysis) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Open mapping theorem (functional analysis)

In research: Open mapping theorem (functional analysis) appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Open mapping theorem (functional analysis) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Open mapping theorem (functional analysis) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in functional analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Open mapping theorem (functional analysis) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Open mapping theorem (functional analysis) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Open mapping theorem (functional analysis) means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Open mapping theorem (functional analysis) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Open mapping theorem (functional analysis) in simple terms?

Why does Open mapping theorem (functional analysis) matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Open mapping theorem (functional analysis)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Open mapping theorem (functional analysis).