In order to define any mathematical objects of significance, the following definitions are required:
Set - An unordered collection of unique objects. This is arguably the foundational object for all of mathematics (although one can make a similar argument for categories - don't worry about that). Expressed with the notation
{a, b, c}
to describe a set consisting of elements a,b,c. May contain finitely or infinitely many elements.
n-Tuple - An ordered collection of n objects, where n is generally finite. Although a rigorous set-theoretic definition exists, it is convenient to simply think of an n-tuple as a primitive object in its own right. Denoted
(x1, x2, ..., xn)
for a tuple of n elements.
Ordered Pair - A 2-tuple. Naming convention can be extended to "ordered triple", "ordered quadruple", etc.
From here, things get somewhat convoluted, for there are at least two ways to build more complicated structures: One via Topology and one via Abstract Algebra (I suppose a third can arise through order theory). Since the algebraic approach is more accessible and easily described, I will start with these.
Magma - An ordered pair (X, ∙), where X is a set and ∙, referred to as multiplication, is a binary operation closed under X.
Semigroup - An magma in which the multiplication operation ∙ is associative.
Monoid - A semigroup in which there exists an identity element, which is denoted 1. For any element x in X:
x∙1 = 1∙x = x
Group - A monoid in which every element x has a multiplicative inverse. For any element x in X there is an element x':
x∙x' = x'∙x = 1
Abelian Group - A group in which the multiplication operation is commutative. In the case of an abelian group, we traditionally use an additive notation rather than a multiplicative one. Thus the group operation is referred to as "addition", the inverse of an element x is denoted (-x), and the identity element 1 is denoted 0. This notational change is in order to remain consistent with the notation of rings (below).
Rng - An ordered triple (X, +, ∙), where X is a set, (X, +) is an abelian group, and (X, ∙) is a semigroup. It is also required that both left and right multiplication distribute over addition for elements x,y,z:
(x+y)z = xz + yz
z(x+y) = zx + zy
Ring - A rng in which there also exists a multiplicative identity element (X is a monoid over multiplication). The ring can be thought of as a "space" in which some form of rudimentary arithmetic can be done. The notion of arithmetic might be considered to be fully developed in the setting of a field (see below).
Commutative Ring - A ring in which the multiplication operation is commutative.
Domain - A ring with no zero divisors: There do not exist nonzero elements a,b such that ab=0 (where, remember, 0 is the additive identity element).
Integral Domain - A domain which is also a commutative ring.
Unique Factorization Domain (UFD) - An integral domain in which every element can be decomposed into a product of prime elements ("prime", in this case, having a more general ring-theoretic definition) which is unique up to a permutation of factors and multiplication by unit elements.
Principle Ideal Domain (PID) - An integral domain in which every ideal is generated by a single element. Every PID is a UFD.
Skew Field - A ring in which every element has a multiplicative inverse.
Field - A commutative ring in which every element has a multiplicative inverse (alternatively, a commutative skew field). Every field is a PID. Intuitively, a field is the minimal algebraic structure in which basic arithmetic holds: We can add, subtract, multiply, and divide as we might expect.
R-Module over a Ring - An ordered quadruple (R, M, +, ∙) in which R is a ring, M is a set which is an abelian group with respect to + (addition) and ∙ (scalar multiplication) is a function which maps RxM into M. Scalar Multiplication must also satisfy the following axioms for all elements r,s in R and x,y in M:
r(x+y) = rx + ry
(r+s)x = rx + sx
r(sx) = (rs)x
1x = x, where 1 is the multiplicative identity element of R
(Note: This is the definition of a so-called "left" module. A "right" module maps from MxR into M and has similar properties for left-multiplication. If R is a commutative ring, then these notions are identical.)
Vector Space - A module in which R is a field. In this case, the elements of M are referred to as vectors.
Algebra over a Ring - An R-module together with a bilinear operation, M-multiplication, which acts on the set M.
Algebra over a Field - An algebra over a ring in which the ring R is a field.
Associative Algebra over a Field - An algebra over a field in which M-multiplication is associative.
At this point, further generalizations require taking a look at topological structures:
Topological Space - An ordered pair (X, T) in which X is a set and T is a collection of subsets of X. It is required that T contain the empty set, X itself, and be closed under arbitrary unions and finite intersections. Topological spaces roughly correspond to the geometric of "separability" between points in space. The subsets of X which belong to T are referred to as open sets and define the fundamental topological (and geometric) structure of the space. An open set which contains a point x is referred to as a neighborhood of X.
Kolmogorov Space (T0) - A topological space which has the property that for any two points a,b in X, either a or b has a neighborhood which does not contain the other point.
Frechet Space (T1) - A topological space which has the property that for any two points a,b in X, both a and b have neighborhoods which do not contain the other point.
Hausdorff Space (T2) - A topological space which has the property that for any two points a,b in X, a and b possess neighborhoods which are disjoint from one another.
Metric Space - An ordered pair (X, d) in which X is a set and a "distance function" d from X into the positive real numbers with the following properties for any elements x,y,z in X:
- d(x,y) > 0 if x != y (Non-Negativity)
- d(x,y) = 0 iff x = y
- d(x,y) = 0 iff x = y
- d(x,y) = d(y,x) (Symmetry)
- d(x,z) <= d(x,y) + d(y,z) (Triangle Inequality)
The distance function allows for a geometric notion of distance to be computed between elements as one might expect. The distance function (also called a metric) induces a topology on X.
Complete Metric Space - A metric space in which every Cauchy sequence converges. Intuitively, all sequences of elements in the space which grow increasingly close together converge to a point which is indeed an element of the space.
At this point, the algebraic and topological notions come together.
Topological Vector Space - A vector space which also has a topology defined on it.
Normed Vector Space - A vector space, which is over some a subfield of the complex numbers, together with a function p which sends a vector into the positive real numbers and has the following properties for all a in the field F and vectors x,y:
- p(ax) = |a|p(x)
- p(x+y) <= p(x) + p(y) (Triangle Inequality)
- p(x) = 0 iff x is the zero vector
The similarity with the definition of a metric is not accidental, as the norm induces a metric, and thus a topology, on the vector space.
Banach Space - A normed vector space which is also a complete metric space.
Inner Product Space - A vector space, which is over either the real or complex numbers, together with a binary function <∙,∙>, the inner product, which sends a pair of vectors into the positive real numbers and has the following properties for all a in the field F and vectors x,y,z:
- <x,y> = <y,x>*, where * denotes the complex conjugate
- <ax,y> = a<x,y>
- <x,x> >= 0, where equality holds only if x = 0
The inner product induces a norm, and thus a metric and a topology, on the vector space. Intuitively, the inner product space extends the normed vector space by introducing enough structure that the notion of angle can be defined; indeed, any normed vector space whose norm satisfies the parallelogram law can be extended to be an inner product space.
Hilbert Space - An inner product space which is also a complete metric space. Note that every Hilbert Space is also a Banach Space.
That's all I've got for now...I'll update this on occasion and maybe put up a post to let anybody who cares know when I do. Let me know if there are any mistakes and I'll be sure to correct them.