 # Metric Spaces Flashcards Preview

## Complex Analysis - Michaelmas > Metric Spaces > Flashcards

Flashcards in Metric Spaces Deck (74):
1

## Define a Metric Space. 2

## What is the formula for the Euclidean norm on ℝn or ℂn? 3

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## What is the formula for the Euclidean norm on ℂ2? 6

## What is the formula for the Euclidean norm on ℝ2? 7

## What is the formula for a Metric induced from inner products in vector spcaes? 8

## What inequality do you use to check the following metric satisfies the triangle inquality? 9

## Define a norm. 10

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## What is the formula for the lp-norm? 13

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## What is the formula for the l∞-norm? 15

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## What is the Riemannian metric? 18

## What is the Discrete metric 19

## What is the subspace metric? 20

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## What is the sunflower metric? 22

## Prove the following sunflower metric satisfies (D3) the triangle inequality.  23

## What is the 'jungle river' metric? 24

## Prove the following 'jungle river' metric is a metric. 25

## Define open and closed balls in metric spaces. 26

## Draw the unit balls for the l1, l2, l∞. What does the lp-norm look like in comparison? 27

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## Define open/closed sets in a metric space. 31

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## What is a lemma about open balls? 34

## Prove the following Lemma.  35

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## What is the generic formula for a ball with respect to the discrete metric space? 37

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## Prove balls with respect to the discrete metric are clopen. 39

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## What is the Lemma about the union/intersection of open sets? 42

## Prove the following Lemma.  43

## What is the corolllary about closed sets to the following Lemma.  44

## Let A be a subset of a metric space (X,d). Define the interior A0 of A. 45

## Let A be a subset of a metric space (X,d). Define the closure Ā of A. 46

## Let A be a subset of a metric space (X,d). Define the boundary ∂A of A. 47

## Let A be a subset of a metric space (X,d). Define the exterior Ae of A. 48

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## Finish the following two statements about the properites of a subset A ⊂ X?  50

## Define limits and convergence in a metric space. 51

## Finish the following Lemma.  52

## Prove part (i) of the following Lemma.  53

## Prove part (ii) of the following theorem.  54

55

## Define continuity. 56

## What is the Lemma about the basic properties of continuous functions? 57

## What is a preimage? 58

## What is the theorem about continuity and open/closed sets. 59

## Define compact. 60

## What is the Lemma about convergence of subsequences? 61

## Prove the following Lemma.  62

## What is the proposition about clsoed sets and limits of sequences? 63

## What is the corollary about closed sets vs. closedness? 64

## Prove the following.  65

## Prove the following.  66

## Prove the following.  67

## Define bounded. 68

## What is the Lemma about compact sets and boundedness? 69

## Prove the following Lemma.  70

## What is the Heine-Borel for ℂ theorem? 71

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## Finish the following Lemma.  73

## What is the theorem about the continuous image of a compact set being compact? 74

## Prove the following theorem.  