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locally compact – metric space
locally compact
A topological space is locally compact if every point of the space has a neighborhood whose closure is compact.
locally integrable
A measurable function on a finite-dimensional real space with range in the complex numbers is said to be locally integrable if for all compact sets K we have:
The space of all such functions is denoted L1LOC.
Cf. Lebesgue integral.
locus
A set of points satisfying a (geometric) condition. E.g., a parabola is the locus of points equidistant from a given point and a given line.
log
See logarithm.
logarithm
A logarithm is an expression of the form log bA, where b is called the base of the logarithm and A is called the argument, and it evaluates as the exponent to which the base must be raised to return the argument. Logarithms may be viewed as an alternative form of exponential notation, since we have log bA = x if and only if bx = A. A common logarithm is a logarith with base 10, and a natural logarithm (often written “ln”) is a logarithm with base e.
Cf. laws of logarithms, Euler number.
logarithmic function
A function which acts on its argument with a logarithm.
logic
In the broadest sense, the term “logic” refers to the reasoning processes or forms of argument people use to reach valid conclusions from one or more premises (assumptions). More specifically, any particular logic is a system of inference, by which one reasons from premises to conclusions, or by which one validates one’s conclusions. An incorrect inference, i.e., one which violates the rules of logic, is called a fallacy, and conclusions reached by means of a fallacy are called invalid. (By contrast, when the premises of an argument are false but the argument is formally valid, then the conclusion is called unsound.)
Ordinary logic falls into two broad types; deductive and inductive. Reasoning from general premises to a specific conclusion is deductive. For example, knowing that heavy clouds and a brisk breeze often signal an oncoming storm, one could conclude deductively that it might rain this afternoon. Reasoning from specific premises to a general conclusion, on the other hand, is inductive. For example, observing that every raven one sees is black, one might conclude inductively that all ravens are black. Deductive conclusions are necessarily true if the premises are true and the logic used formally valid (free of fallacy). Inductive conclusions are never certain, but are only more or less reliable. In mathematics only deductive reasoning is used (but see the entry for mathematical induction).
Mathematicians use symbolic logic, that is, logic that can be reduced to a purely syntactical system which disregards the semantic content of any of the symbols in use. The two primary forms of symbolic logic are the propositional calculus and the predicate calculus. The latter augments the former with existential and universal quantifiers (permitting sentences that begin with “there exists ___ such that” and “for all ___”). When quantification is permitted only over elements of the universe of discourse, the logic is called “first-order.” When quantification over classes of objects and/or over predicates is permitted, it is called second-order logic.
Cf. axiom, fuzzy logic, equational logic, syllogism, Hempel's Ravens Paradox.
lower bound
A lower bound of an ordered set is an element which is less than or equal to every element in the set.
Cf. greatest lower bound.
l.u.b
Abbreviation for least upper bound.
Mahlo cardinal
A cardinal k is called Mahlo if it is inaccessible and {a < k:a is inaccessible} is stationary in k.
matrix
A rectangular array of, usually, real or complex numbers, organized into rows and columns. When specifying the size of a matrix, the number of rows is stated first. For example, here is a 2 by 4 real matrix:
The numbers in a matrix are called entries, and are specified by the row and then column in which they appear. In the example above, for instance, A1,2 is the entry in the first row and second column, i.e. 4.98.
Two matrices of the same size (with the same number of rows and columns) can be added componentwise, that is (A + B)i, j = Ai, j + Bi, j. Matrix multiplication is a little more complicated. A matrix M can be multiplied on the right by another matrix T only if T has the same number of rows as M has columns, and the result will have as many rows as M and as many columns as T. The entry of M T appearing in row i and column j is the sum of the pairwise products of the entries in row i of M and column j of T. In other words, if M is an n by m matrix, T must be an m by l matrix, and
Notice that this operation is not, in general, commutative.
A matrix that has the same number of rows and columns is called a square matrix, and these are particularly interesting as they can be added and multiplied, and the results will have the same dimensions.
The most standard use of matrices is to represent linear operators, but they have a wide variety of other uses, and are interesting to study in their own right. Matrices can have a large number of different types of entries, including various kinds of numbers, elements of abstract fields and other algebraic structures, functions, and so forth.
maximal filter
See filter.
measurable cardinal
If there exists a non-principal k-complete ultrafilter on a set of size k, then k is called a measurable cardinal.
measurable function
Given measure spaces (X, M, m) and (Y, N, n), a function f from X to Y is said to be measurable if the inverse image of every measurable set in Y is measurable in X.
measurable space
A set X together with a s-algebra of sets M defined on it is called a measurable space, usually denoted (X, M). The elements of M are called measurable sets.
Cf. measure.
measure
Given a set X and a s-algebra of sets M defined on X, a measure on X is a positive real-valued function m whose domain is M, and which satisfies: - the measure of the empty set is zero.
- (Countable additivity) For any countable, disjoint sequence of sets in the domain of m, the measure of the union of the sequence is equal to the sum of the measures of the sets in the sequence.
If the measure of every set in M is finite, then m is called a finite measure. If X can be expressed as a union of sets in M all of whose measures are finite, then m is called a s-finite measure. If for every set E in the domain of m whose measure is infinite there is a subset F of E which has finite measure, then m is called a semi-finite measure. If for any set E in M which has zero measure all the subsets of E are also in M (and therefore have zero measure), then m is called a complete measure.
Cf. measure space, signed measure, outer measure, Lebesgue measure, Borel measure.
measure space
A set X together with a s-algebra of sets M defined on it and a measure m defined on M is called a measure space, usually denoted by (X, M, m).
meet
A binary operation whose value on two elements a and b of a lattice is the greatest lower bound of a and b, denoted a  b.
Cf. join.
meet-morphism
A function f on a lattice L is called a meet-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a meet-morphism, then f is called a meet-isomorphism. A meet-isomorphism from a lattice to itself is called a meet-automorphism.
Cf. meet, join-morphism.
metric
Given a set X, a metric on X is a function d with domain X and range in R* (the set of non-negative real numbers) satisfying, for all a, b, and c in X:- d(a,b) = 0 if and only if a = b;
- d(a,b) = d(b,a); and
- d(a,b) + d(b,c) >= d(a,c
(where >= means ‘greater than or equal to’).
Cf. metric space.
metric space
A set with a metric defined on its elements.
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