Definitions Flashcards
Banach Space
A complete metric space.
Measurable Set
A subset E of a real number vector that is said to be Lebesgue measurable, or simply measurable, if given e>0, there exists an open set G such that. E is a subset of G and ||E - G||<e
Cauchy Sequence
A sequence of numbers where successive terms become closer together as the list progresses.
Disjoint
A term where two sets share no common element(s)
Hilbert Space
A complete Inner Product Space
Condition of a Cauchy Sequence
In a sequence {a_n} with length n, if for every real number e, there exists a positive integer N such that for all natural number m and n within N, the absolute difference between any 2 terms can be defined as |a_m - a_n| < e
Normed Linear Space
A real/complex vector space in which every vector is associated with a real number, called its absolute value or norm.
Linear Operator
A mapping/function acting on the elements of a vector space while preserving its linear structure.
Supernum
Least upper bound
Infernun
Greatest lower bound
Metric Space Propeties
1) d(x,y) = d(y,x)
2) d(x,y) >= 0 & d(x, y) = 0 <=> x = y
3) d(x,y) <= d(x,z) + d(z, y)
A subset is dense in a set under what condition.
A set E being a subset of E1 is dense in E1 if for every x1 being an element of E1 and e>0, there is a point x being an element of E such that 0<|x-x1|<e. If E=E1, then E is dense in itself.
Dense in Topology
A subset D of a space X is dense if every point in X is either in D or at least close to a point in D. Ex: all rational numbers Q are dense in all real numbers R.
Neighborhood
In topology, a neighborhood of a point x in a space X is any set that contains an open set around x. Meaning, a set N is a neighborhood of x if there exists an open set U such that x is an element of U which is a subset of N.
Lebesgue Measurable Function
A function f defined on a measurable domain E that maps into extended real numbers. It’s Lebesgue Measurable if for every real number c, the set x within E and f(x) > c is measurable.
Sigma Algebra
A function between measurable spaces that preserves the structure of the spaces
Riemann Integrable
A function that can be calculated by calculating the limit of Riemann sums as the partition of the interval becomes finer in a closed interval [a,b].
Riemann Sums
An approximation of an area under the curve, calculated by dividing the region into smaller shapes, particularly rectangles grouped together in an interval [a,b].
Compliment of a set
A set such that Ac contains the opposite conditions of the original set A.
Isometry
A transformation between metric spaces that preserves distance.
What Makes a Metric Space Complete?
A metric space is … if for every Cauchy Sequence in that space converges to a point within the space.
Hamel Basis
A set of linearly independent vectors in a vector space, such that every element of the space can be expressed as a finite linear combination of these basis vectors.
Lp Spaces
Function Spaces defined using the p-norm
Lebesgue Integral
A method of integrating that subdivides the area under a curve with horizontal rectangles.
Fixed point
A point x in a mapping T: X-> X such that Tx = x. Meaning, the point remains in its original location no matter how much the space contracts, reflects, or rotates.
Contraction
A mapping that reduces the size of a metric space X such that d(Tx, Ty) <= a*d(x,y) for all x, y within X and a<1.
