Maths WorkshopMaths problems with full solutions

Metric and normed spaces

21 problems · Topological Properties, Convergence, Compactness and Completeness.

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Ex.254Metric spaces

#metric #definition #axioms

Metric and normed spaces

Ex.255Open balls in a metric space

#open ball #metric #definition

Metric and normed spaces

Ex.512Cauchy sequences

#Cauchy #sequences #metric

Metric and normed spaces

Ex.513Continuity in metric spaces

#continuity #epsilon-delta #metric

Metric and normed spaces

Ex.514Convergence in metric spaces

#convergence #sequences #metric

Metric and normed spaces

Ex.515Classical metrics on $\mathbb{R}^n$

#Euclidean #Manhattan #sup norm

Metric and normed spaces

Ex.516Normed spaces

#norm #definition #axioms

Metric and normed spaces

Ex.1062Compactness in metric spaces

#compactness #open cover #sequential

Metric and normed spaces

Ex.1063Compact sets are closed and bounded

#compact #closed #bounded #metric

Metric and normed spaces

Ex.1064Continuous image of compact

#compact image #continuity #compactness

Metric and normed spaces

Ex.1065Complete metric spaces

#completeness #Cauchy sequences #examples

Metric and normed spaces

Ex.1066Equivalent metrics

#equivalent metrics #same topology #norms

Metric and normed spaces

Ex.1067Isometries between metric spaces

#isometry #metric #distance-preserving

Metric and normed spaces

Ex.1068Equivalent norms

#equivalent norms #topology #finite dimension

Metric and normed spaces

Ex.1069Classical normed spaces

#normed spaces #Banach #examples

Metric and normed spaces

Ex.1070Product metrics

#product spaces #metric #sup

Metric and normed spaces

Ex.1779Baire category theorem

#Baire #category #complete

Metric and normed spaces

Ex.1780Completion of a metric space

#completion #Cauchy #dense embedding

Metric and normed spaces

Ex.1781Closed bounded sets need not be compact

#counterexample #compactness #infinite dimension

Metric and normed spaces

Ex.1782Heine–Cantor theorem

#Heine-Cantor #uniform continuity #compact

Metric and normed spaces

Ex.2182Banach fixed point theorem

#Banach #fixed point #contraction

Metric and normed spaces