Real Analysis Flashcards
What makes a function ρ a metric?
1) ρ(x,y) ≥ 0 and equals 0 iff x=y
2) ρ(x,y)=ρ(y,x)
3) ρ(x,z) ≤ ρ(x,y) + ρ(y,z)
What is “a set is open” equivalent to?
It is a union of balls
What is the interior of A, int(A), defined as?
The largest open set contained in A
What is int(A) equal to?
The union of all open sets in A
When is x a limit/cluster/accumulation point of a set A? (x in X, A a subset of X)
If every ball about x contains a point of A other than x
How is the set of limit points in A denoted?
A’
What is the relation between A being closed and its limit points?
It is closed if it contains all of its limit points.
A set A is closed if and only if?
Every convergent sequence in A has its limit in A
If A is closed, what does this mean about its compliment?
The compliment of A is open
What is the compliment of A? (X\A)
The set of all points in X which don’t belong to A
What is the closure of A?
It is the smallest closed set containing A
What is the closure of A, cl(A) equal to?
The union of A and A’, or the intersection of all the closed sets containing A.
Which three statements are equivalent for a linear map T?
1) T is continuous
2) T is continuous at 0
3) T is bounded
If a Cauchy sequence has a convergent subsequence, what does the subsequence converge to?
The same limit
When is a metric space complete?
If every Cauchy sequence converges
If (X,ρ) is complete and A is a subset of X, what is the completion of (A,ρ)?
(cl(A),ρ)
What type of space is a complete normed linear space?
A Banach space
How many fixed points does a contraction have and is it continuous?
A contraction is continuous and has at most one fixed point.
What do we say about A if no disconnection of it exists?
It is connected
Is the interval [0,1] connected?
Yes
When is a set A clopen?
If it’s both closed and open
What is the relationship between a subset A of X being connected, and clopeness?
A is connected if (A,ρ) has no non-trivial clopen set
When is a subset K of X sequentially compact?
If every sequence in K has a convergent subsequence whose limit lies in K
What is a cover called if every member of it is open?
An open cover
What is a cover of K?
A collection of subsets for which K is the subset of the union of
When is a subcollection of a cover called a subcover?
When it is also a cover of K, but smaller than the cover it is a subcollection of.
When is K compact? (Heine Borel property)
If every open cover of K admits a finite subcover
What is a finite cover?
A cover that has finitely many members
In a metric space, what is sequential compactness equivalent to?
Compactness
A compact set has what two properties?
It is closed and bounded
What is a property of a closed subset of a compact set?
It is compact
When can we say that (X,ρ) is a compact space?
If X is a compact set
A set in R^n is totally bounded if and only if it is?
Bounded
A metric space is compact if and only if?
It is complete and totally bounded
A continuous function on a compact space is?
Uniformly continuous
If T is a contraction on a complete space what does it have?
A unique fixed point
