# Tag Archives: analysis

## Commutative Algebra 57

Continuing from the previous article, A denotes a noetherian ring and all A-modules are finitely generated. As before all completions are taken to be -stable for a fixed ideal . Noetherianness We wish to prove that the -adic completion of … Continue reading

Posted in Advanced Algebra | | 1 Comment

## Topology: Basic Definitions

Motivation and Definition While studying analysis, one notices that many important concepts can be defined in terms of “open sets”. One gets the inkling that this concept is critical in forming our notions of continuity, limits etc. In this article, we … Continue reading

## Basic Analysis: Closed Subsets and Uniform Continuity

Let’s consider another question: suppose f : D → R is continuous, where D is a subset of R. If (xn) is a sequence in D converging to some real L, is it true that (f(xn)) is also convergent? Now if L is in D, then we know that (f(xn)) → (f(L)). … Continue reading

## Basic Analysis: Uniform Convergence

Once again, let be a subset. Suppose we now have a sequence of functions , where n = 1, 2, 3, … , such that for each x in D, the sequence converges to some real value. We’ll denote this value by f(x), thus … Continue reading

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## Basic Analysis: Differentiation (2)

Finding Extremum Points One of the most common applications of differentiation is in finding all local maximum and minimum points. Definition. We say f(x) has a local maximum (resp. minimum) at x=a, if there is an open interval (b, c) containing a, such … Continue reading

## Basic Analysis: Differentiation (1)

In this article, we’ll look at differentiation more rigourously and carefully. Throughout this article, we suppose f is a real-valued function defined on an open interval (b, c) containing a, i.e. f : (b, c) → R with b < a < c. Theorem. The derivative of f(x) at a is … Continue reading

## Basic Analysis: Limits and Continuity (3)

Let’s consider multivariate functions where . To that end, we need the Euclidean distance function on Rn. If x = (x1, x2, …, xn) is in Rn, we define: Note that |x| = 0 if and only if x is the zero vector 0. Now we are ready … Continue reading

## Basic Analysis: Limits and Continuity (2)

Previously, we defined continuous limits and proved some basic properties. Here, we’ll try to port over more results from the case of limits of sequences. Monotone Convergence Theorem. If f(x) is increasing on the open interval (c, a) and has … Continue reading