Finite-dimensional normed spaces are complete
Builds on Equivalence of norms in finite dimensionBanach spaceComplete metric space
Statement
Let be a finite-dimensional vector space over or , with a norm . Then is a Banach space.
Remarks
Proof: by equivalence of norms in finite dimension, the norm is equivalent to the sup norm of the coordinates in a basis. Equivalent norms have the same Cauchy sequences and the same convergent sequences. With the sup norm, completeness follows from the completeness of in each coordinate. A consequence: each finite-dimensional subspace of a normed space is complete, and thus closed.
Reading path
To understand this page from the ground up, read these 56 pages in order. The foundations come first.
- Element definition
- Euclidean metric example
- Extended real number line object
- Proposition concept
- Real numbers object
- Set bounded above definition
- p-metric on R^n example
- Set definition
- Conjunction definition
- Disjunction definition
- Implication definition
- Predicate definition
- Supremum definition
- p-norm on R^n example
- Empty set object
- Family of sets definition
- Ordered pair definition
- Intersection definition
- Union definition
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- Union of a family of sets definition
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Topological space definition
- Complement definition
- Function definition
- Subspace topology definition
- Preimage of a set definition
- Metric space definition
- Closed ball definition
- Convergence of a sequence in a metric space definition
- Open ball definition
- Diameter of a set definition
- Bounded set definition
- Metric topology definition
- Epsilon characterisation of Cauchy sequences statement
- Complete metric space definition
- Open set definition
- Closed set definition
- Continuous map of topological spaces definition
- Equivalent metrics definition
- Open cover definition
- Equivalent norms definition
- Compact space definition
- Banach space definition
- Compact set definition
- Heine–Borel theorem statement
- Compactness via open covers statement
- Continuous image of a compact set is compact statement
- Continuous real function on a compact set attains its bounds statement
- Equivalence of norms in finite dimension statement
- Finite-dimensional normed spaces are complete statement
Where this leads
1 page builds on this one.
Further reading
- Web https://en.wikipedia.org/wiki/Banach_space, Complete norms and equivalent norms
About this page
| Tag | 0ABP2F2ZP7 |
|---|---|
| URI | m2://0ABP2F2ZP7 |
| Kind | statement, theorem |
| Writing | W0 stub |
| MSC | 46B99 |
| Status | This copy of the site has no build data. Run m2 status 0ABP2F2ZP7. |
| Uses | Equivalence of norms in finite dimension, Banach space, Complete metric space |
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