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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
Tagged a-adic filtrations, algebraic geometry, analysis, completion, filtrations, hensels lemma, local rings, p-adic
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
Posted in Notes
Tagged analysis, closed subsets, discrete topology, metric spaces, open balls, open subsets, topology, topoloical equivalence
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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
Posted in Notes
Tagged advanced, analysis, closed subsets, continuity, points of accumulation, uniform continuity, uniform convergence
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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
Posted in Notes
Tagged advanced, analysis, continuity, convergence, pointwise convergence, series, uniform convergence
1 Comment
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
Posted in Notes
Tagged advanced, analysis, compact sets, differentiation, mean value theorem, rolle's theorem
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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
Posted in Notes
Tagged advanced, analysis, analytic, chain rule, differentiation, taylor series
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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
Posted in Notes
Tagged advanced, analysis, continuity, limits, multivariate, open balls, open subsets, topology
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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
Posted in Notes
Tagged advanced, analysis, continuity, limits, monotone convergence theorem, points of accumulation, squeeze theorem
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Basic Analysis: Limits and Continuity (1)
[ This is a continuation of the series on Basic Analysis: Sequence Convergence. ] In this article, we’ll describe rigourously what it means to say things like . First, we define a punctured neighbourhood of a real number a to be … Continue reading
Basic Analysis: Sequence Convergence (4)
In this article, we’ll consider the convergence of an infinite sum: . We call this sum an infinite series. Let be the partial sums of the series. Definition. We say that is L (resp. ∞, -∞) if the partial sums converge to … Continue reading
Posted in Notes
Tagged abel transformation, alternating series, analysis, basel series, convergence, series, telescoping series
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Mathematics and Such 