Theorems Flashcards
What is the definition of a normed vector space?
A normed vector space is a vector space equipped with a norm, which is a function that assigns a non-negative value to each vector and satisfies certain properties.
True or False: The triangle inequality holds in a normed vector space.
True
In a normed vector space, what property does the norm satisfy with respect to scalar multiplication?
The norm satisfies the property that the norm of a scalar multiple of a vector is equal to the absolute value of the scalar multiplied by the norm of the vector.
What is the Cauchy-Schwarz Inequality in a normed vector space?
The Cauchy-Schwarz Inequality states that for any vectors u and v in a normed vector space, the absolute value of their inner product is less than or equal to the product of their norms.
What is the definition of completeness in a normed vector space?
Completeness in a normed vector space means that every Cauchy sequence in the space converges to a limit that is also in the space.
Banach Space
A complete, normed vector space, meaning it’s equipped with a norm that allows for the measurement of vector length and complete in the sense that every Cauchy Sequence within the space converges to a limit.
Bounded linear space theorem
The theorem often refers to the equivalence between boundedness and continuity of linear operators in normed spaces. Specifically, a linear operator between normed spaces is bounded if and only if it is continuous.
Linear Operator theorem
This theorem states the equivalence of boundedness and continuity for linear operators between normed spaces. This theorem states that a linear operator between two normed spaces is bounded if and only if it is continuous.
Properties of an operator norm
1) non-negativity and definiteness
2) scalar multiplication
3) triangle inequality
(Proceed to explain all 3 properties)
Subspace theorem
For a system of linear inequalities involving linear forms in several variables, the solutions must lie in a finite number of proper subspaces. Specifically, if the solutions to these inequalities are bounded in a certain way, they are constrained to a limited number of lower-dimensional subspaces in a vector space.
Baire Category Theorem
If M is a complete Metric Space and a collection of Closed Subsets C1, …, Cn exists such that M is the union of all the subsets, then at least 1 closed subset contains an open Ball B. (i.e. a closed subset has an interior point)
Closed Subsets vs. Open Subsets
A closed subset in topology includes boundary points. An open subset doesn’t.
Complete Metric Space
A metric space in which every Cauchy sequence converges to a limit within the space itself. Meaning, it contains no “missing points” where a sequence could converge outside the space.
Uniform Boundedness Theorem
For a family of continuous linear operators acting on a Banach space, if each operator is pointwise bounded on the space, then the operators are uniformly bounded on every bounded subset of the space. This means there is a single bound that applies to all operators in the family across the space.
Open Mapping Theorem
A fundamental result in functional analysis stating that if a continuous linear operator between Banach spaces is surjective, then it is an open map.
Closed Graph Theorem
For a linear operator between two Banach spaces (or more generally, Fréchet spaces), the operator is continuous if and only if its graph is closed in the product space.
Zorn’s Lemma
If a partially ordered set has the property that every chain (i.e., totally ordered subset) has an upper bound, then the set contains at least one maximal element.
Hamel Basis
A set of vectors in a vector space such that every vector in the space can be uniquely expressed as a finite linear combination of these basis vectors.
Hamel Basis Theorem
If V is a vector space, then it has a Hamel Basis.
Dual Space Theorem
If a normed space V exists, then for all non-zero elements v within V, there exists a function for within V’ such that ||f||=1 and f(v)=||v||.
Can a Banach Space be reflexive?
Yes, if it is isomorphic to its double dual, meaning each element of the space corresponds uniquely to an element of its double dual. Such a space has the property that every bounded sequence contains a weakly convergent subsequence.
Lebesgue Measure Theorem
A function f is Lebesgue Integrable if its Lebesgue Measure is finite.
Outer Measure Theorem
A fundamental result in measure theory that provides a method to construct a measure from an outer measure. An outer measure is a function defined on all subsets of a given set, taking values in the extended real numbers and satisfying specific properties: it assigns zero to the empty set, is countably subadditive, and is monotonic.
What does UROHC stand for in proofs
Understand what to prove
Recall the necessary information you need
Outline what to understand
Hydrate the proof (fill if you recall, put a placeholder if you don’t)
Check whether the proof is complete and rehydrate where needed
UROHC
Uncle Roger Owns Happy Cows
ZST
Zeus Scares Tigers
What does ZST stand for in properties of a norm?
1) Zero Vector condition
2) Scalar Vector condition
3) Triangle Inequality
Linear Independence in Applied Analysis
For each scalar (a_n) and vector (v_n) in a sequence of a finite length n, the sum of all products (a_n*v_n) is nonzero. However, the series is zero if and only if all the scalars a_1 to a_n is equal to zero.
Open Set Theorem
1) Any union of open sets is also open
2) The intersection of a finite number of open sets is open
Closed Set Theorem
1) The intersection of any number of finite closed sets is closed
2) The union of a finite number of closed sets is closed
Heine-Borel Theorem
A set of real vectors E is compact if and only if E is closed and bounded
Closed Set Theorem
A set of real vectors E (R^n) is compact if and only if every sequence of points of E has a subsequence that converges to a point of E.
Monotone Convergence Theorem
If a sequence of functions {fn}n is a sequence in L+(E) such that f1<=f2<=… on E and fn —-> f pointwise on E (lim n-> infinity fn(x) = f(x)), then lim n-> infinity lebesgue integral of fn = lebesgue integral of f.
DeMorgan’s Laws
- The compliment of the union of two sets A and B is equal to the intersection between A and B
- The compliment of the intersection between two sets A and B is equal to the union of the sets A and B.
Dominated Convergence Theorem
If a sequence of measurable functions f_n converges pointwise to a function f and is dominated by an integral function g, then:
1) f is integrable
2) f_n and the Lebesgue integral of f_n approach both f and the Lebesgue integral of f as n approaches infinity
Banach Fixed Point Theorem
Consider a metric space X = (X, d) where X is not empty. If X is complete and let T: X -> X be a contraction on X. Then T has precisely one fixed point.
Continuity Lemma
A contraction T on a metric space X is a continuous mapping.
What does the Monotone Convergence Theorem state?
If a sequence of measurable functions is monotonically increasing and converges pointwise to a limit, then the integral of the limit equals the limit of the integrals.
True or False: The Monotone Convergence Theorem applies to non-negative functions only.
True
Fill in the blank: The Bounded Inverse Theorem states that if a linear operator is _____ and _____, then it has a bounded inverse.
bounded, bijective
What is a necessary condition for the Bounded Inverse Theorem to hold?
The operator must be continuous and defined on a Banach space.
Multiple Choice: Which of the following is a key property of a compact operator?
It maps bounded sets to relatively compact sets.
What is the significance of the Uniform Boundedness Principle in Functional Analysis?
It states that for a family of continuous linear operators, pointwise boundedness implies uniform boundedness.
True or False: The Riesz Representation Theorem provides a way to represent continuous linear functionals on a Hilbert space.
True
What does the Hahn-Banach Theorem allow us to do?
It allows the extension of bounded linear functionals.
Fill in the blank: In the context of the Monotone Convergence Theorem, the sequence of functions must be _____ to ensure the theorem applies.
measurable
What type of convergence is primarily addressed by the Dominated Convergence Theorem?
Almost everywhere convergence of functions.
What is the main statement of the Hahn-Banach Theorem?
The Hahn-Banach Theorem states that if a linear functional is bounded on a subspace of a normed space, it can be extended to the whole space without increasing its norm.
True or False: The Hahn-Banach Theorem only applies to finite-dimensional spaces.
False
Fill in the blank: The Hahn-Banach Theorem is a fundamental result in the field of _____ theory.
functional analysis
Which property of linear functionals is preserved when applying the Hahn-Banach extension?
The norm of the functional is preserved.
Multiple Choice: Which of the following is a consequence of the Hahn-Banach Theorem? A) Every normed space is complete. B) Every bounded linear functional can be represented as an inner product. C) There exists a continuous linear functional that separates points from closed convex sets.
C) There exists a continuous linear functional that separates points from closed convex sets.
