Analysis is a somewhat vague term used to mean that the methods under discussion involve objects that except in special cases may not be computed with finite algorithms. In the first two years of undergraduate mathematics in the United States any of the following terms indicate that analysis is involved:
limit.
derivative.
integral.
infinite sum.
Of course, finite sums are algorithmic special cases of infinite sums. In polynomial contexts limits, derivatives, and usually integrals (certainly one-dimensional integrals) are algorithmic.
Basic analysis is the study of the minimal ``closed'', (more formally ``complete'') contexts for analysis where one insists that arithmetic with the (ordinary) integers be available.
All such contexts were classified about one hundred years ago, and today it is well known that there are such contexts other than that of the real numbers. All of the contexts other than that of the real numbers are called non-Archimedean.
There is a one-to-one correspondence between the set of positive integers that are prime and the set of (equivalence classes of) non-Archimedean contexts. The non-Archimedean context corresponding to a prime number p is the context of ``p-adic'' numbers.
It is known that these minimal contexts for analysis with the arithmetic of the integers comprise all instances of the abstractly defined notion of minimal non-discrete locally compact topological field of characteristic 0.
The term local field is a modest generalization: this term means ``non-discrete locally compact topological field''.
Every such field admits a unique ``absolute value'' that is characterized by the property that |x| is the ``effect on length'' of multiplying by x.
(The construction of the ``length'' in this rarified atmosphere is that of Haar, which is intrinsic apart from a positive real scalar constant that does not matter for the ``effect on length''.)
An example of a theorem in basic analysis is that every locally compact vector space over a local field must be finite-dimensional. The only way of ``topologizing'' such a vector space is the classical Cartesian way relative to a basis.
This is the view of basic analysis laid out in the first chapter of André Weil's (1906-1998) book *Basic Number Theory* (Springer, 1967), which is a rather advanced book on a different subject that is pitched to the level of second year graduate education in the United States.
While analysts probably should feel bad about the amount of time that it seems to be taking for this to be filtered into the curriculum, algebraists should feel likewise about the computationally relevant and theoretically important notion of ``Gröbner basis'' and geometers should feel likewise about the computationally relevant and theoretically essential notion of ``functor of points''.