Math Flashcards
Define span S
Definition 34:
Let S be a subset of V. The subspace generated by S, denoted by span S, is the smallest vector subspace of V containing S. It coincides with the intersection of all vector subspaces of V
Define a basis
Definition 40:
A set of vectors S of V is said to be a basis of V is S is a linearly independent set such that span S = V
Define dimension
Definition 46:
A vector space is said to have finite dimension if it has a basis with a finite number of elements
Definition 52:
The dimension of a finite dimensional vector space V is the number of elements of a basis of V
Define a linear functional
Definition 57:
A function L that maps from a vector space to real values is called a functional. It is linear if it satisfies the ‘decomposition’ property.
Define dual space of linear functionals
Definition 65:
The set of all linear functionals defined on a vector space V is called dual space of V and is denoted by V’
What is the relation between dimension of vector space V and its dual space V’
dim V = dim V’
Define a linear operator
Definition 82:
A function T that maps from V1 to V2 is called an operator. It is linear if it satisfies the ‘decomposition property’
Define the product of two linear operators T and S
(ST)(v) = S(T(v))
What is an isomorphism?
Definition 106:
Two vector spaces V1 and V2 are called isomorphic if there exists a linear bijective operator T. Such operator is called an isomorphism
What is the law of one price?
If two portfolios have the same future payoff across states, they must have the same value
Define when (L, p) satisfies the Law of One Price
If for all portfolios x, x’ in R^n,
R(x) = R(x’) => v(x) = v(x’)
Define a metric space
Definition 138: A space X is called a metric if there exists a positive function d: X x X --> [0, +inf), called distance (or metric), such that, for each x, y, z belonging to X, i) d(x,y) = 0 iff. x = y ii) d(x,y) = d(y,x) iii) d(x,y) <= d(x,z) + d(z,y)
Define the neighborhood to a point x in X of radius e
B_e(x) = {y in X: d(x,y)
Define an interior point and an isolated point
See page 78
Define a boundary point and an accumulation point
See page 78
Define, for a given set A, its closure
The closure of A is given by the union of A with its set of frontier points
Define a sequence
Definition 177:
A sequence of points {xn} of a metric space X converges to x if for each neighborhood of x, there exists n_epsilon>=1 such that xn is in this neighborhood for each n>=n_epsilon
Define a Cauchy Sequence
See page 92 (extension of definition 177)
What is the term for a metric space in which Cauchy sequences are convergent.
Complete
Is the space R^N with any of the metrics d1, d2, d_inf complete?
Yes by Theorem 196
Is the space of C([0,1]) complete?
Yes under the infinity metric by Theorem 197
But not under the d_1 metric!
Define compactness in calculus
Closed and bounded sets, e.g. interval [a,b]
Define an open cover
An open cover of A is any collection of open sets {Gi} such that A is a subset of the union of all Gi
Define a finite subcover
A finite subcover of A is a finite collection of sets, {G1,..,Gn}, taken from the open cover {Gi} that are still able to cover A so that A is a subset of the finite union of the open sets, Gi
Define a compact space
A metric space X is compact if X itself is a compact set
Are compact metric spaces complete?
Yes (by the Bolzano-Weierstrass property; a subset A of a metric space if compact iff. each sequence of A has at least a subsequence that converges to a point in A)
Define total boundedness
A metric space X is totally bounded, if for each epsilon>0, there exists a finite collection of points xi of X such that X = finite union of all neighborhoods of xi
Define the limit of a function
See page 101
Define continuity of a function
See page 103
Under the discrete metric, are singletons closed or open?
Singletons are then open sets
What is the Heine-Borel theorem on compact sets?
A subset of R^n is compact iff. it is closed and bounded under any metric (d1, d2, d_inf)
