Lecture 1 (August 1, 2023) Plan for the Semester, Schedule, Evaluation Scheme, Motivation for Real Analysis, Motivation for Inverse Function Theorem.
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Lecture 2 (August 3, 2023) Examples of derivatives of functions of one variable using animations, Linear approximation of a function with animation and examples, Revision of derivatives of functions of one variable, Looking at these definitions again and connecting them with linear maps, Definition of a function of several variables, Differentiation and Linear Maps, Finding derivative of a linear map from R^n to R^m. Proving that derivative of a function of several variables is unique.
Lecture 3 (August 7, 2023)
Definition of the operator norm of a linear map A: \R^n --> \R^m, Proof of Theorem 9.7 on page 208 of Rudin's book, that is, proved that \|A\|_{op} < \infty and that A is a uniformly continuous function of \R^n into \R^k.
Chain Rule (Theorem 9.15 on page 214 of Rudin's book).
Lecture 4 (August 8, 2023)
Definitions of partial derivatives and directional derivative of a vector valued function.
Theorem 9.17: If f is differentiable at a point x then all its partial derivatives exist at x, and they determine the linear transformation f'(x) completely.
The Jacobian matrix [f'(x)].
Directional derivative of f in the direction u at x, f'(x) u = D_uf(x) =<\grad f (x), u> for f: R^n --> R.
Recalled the Mean Value Theorem (Theorem 5.19) for function of one variable.
Stated the Mean Value Theorem (Theorem 9.19) for function of several variables. It will be proved in the next class.
Lecture 5 (August 10, 2023)
* Proof of the Mean value Inequality (Theorem 5.19)
* Let A =(a_{ij})_{m X n} be the matrix associated with a linear map T:R^n --> R^m w.r.t. the standard bases of R^n and R^m respectively. Then, ||T||_{op}, the operator norm of T, is less than or equal to (\sum_{i,j} a_{ij}^2)^{1/2}.
Lecture 6 (August 14, 2023)
1) Let A \in L(R^n, R^M). Let (a_{ij})_{m \times n} be the matrix associated with A. Since ||A||_{op} <= \\squareroot( \sum_{i,j} a_{ij}^2), it follows by putting B-A in the place of A that if the matrix entries are continuous functions of a parameter, that is, t |----> a-_{ij}(t) continuous for each i,j then the same is true of A.
More precisely,
If S is a metric space, if a_{11}, ....., a_{mn} are real continuous functions on S, and for each p \in S, A_p is the linear transformation of R^n into R^m whose matrix has entries a_{ij}(p), then the mapping p |------> A_p is a continuous mapping of S into L(R^n, R^m).
2) Definition of a continuously differentiable mapping f:E ---> R^m, where E is an open subset of R^n. (Definition 9.20 on page 219).
3) Proved Theorem 9.21: Suppose f maps an open set E \subset R^n into R^m. Then f \in C^1(E) if and only if the partial derivates D_j f_i exist and are continuous for 1<=i<=m, 1<=j<=n.
4) Defined a contraction map (Definition 9.22) and proved the Contraction Principle (Theorem 9.23): If X is a complete metric space and if \phi is a contraction of X into X, then there exists one and only one x \in X such that \phi(x)=x.
Exercise: Prove that the sequence {x_n} constructed in the proof of the Contraction Principle is a Cauchy sequence in X.
Real Analysis II
Real Analysis II
Lecture 7 (August 17, 2023)
1) Stated the Inverse Function Theorem (one dimensional case) If f is a C^1 function on an interval (a,b) and if f'(a_0) \neq 0 for some a_0 \in (a,b), then f is one -to-one in a neighbourhood of a_0 in (a,b), the function f^{-1} is invertible at b_0 =f(a_0) and (f^{-1})'(b_0) = 1/f'(a_0).
Idea of the proof was given in the class: Without loss of generality we can assume that f'(a_0) >0 and by the continuity of F' we have f' is continuous in a neighbourhood U of a_o in (a,b). Thus f becomes strictly increasing on U.Hence f is one-to-one on U. Let V=f(U). Then, f: U ----> V is a bijection. Let g be the inverse of f, then since f strictly increasing function its inverse g is also a strictly increasing function (prove this). We can prove that g is a continuous function on V. Then we prove that g is differentiable on V by first principle by observing that [g(t)-g(s)]/ (t-s) is the same as the reciprocal of [f(g(t))-f(g(s))]/ [g(t)-g(s)]. Taking limit as t approaches s and a couple of careful steps yields us the proof.
Theorem 14.43 on page 298 of the online notes is another version of the Inverse Function Theorem. But the proof is similar once we assume (without loss of generality) that f'(a_0) >0.
2) Stated the Reflection Theorem (Theorem 14.26 on page 294 in the online notes): Let A and B be two sets. Let f: A ---> B be a function which has an inverse g:B-->A. Then, for all (a,b) \in A X B, (a,b) \in graph(f) if and only if (b,a) \in graph (g).
Remark: If f: (c,d) --> R is a bijective map then the Reflection Theorem says that if g is an inverse of f then graph of g is a reflection of graph of f about the line {y=x}.
The online notes can also be referred to for the definition of inverse of a function, examples of inverse of a function, pictorial illustrations of inverse of a graph and that of the proof of the Reflection Theorem.
3) Proved the Inverse Function Theorem (Theorem 9.24). Explained once again the Application of Inverse Function Theorem to a system of equations discussed in Lecture 1 (as in this file). Inverse function Theorem says that if f is a C^1 function and if f'(x_0) is invertible then f is invertible in a neighbourhood of a_0.
Exercises:
1. If f :(a,b) --> R is a strictly increasing function then inverse of f is strictly increasing.
2. If f :(a,b) --> R is a strictly decreasing function then inverse of f is strictly decreasing.
3. Prove that for an invertible linear transformation, ||A||_{op} \neq 0.
4. Prove Theorem 9.7 (c).
5. Prove Theorem 9.8.
6. In the proof of the Inverse Function Theorem (Theorem 9.24), prove inequality (51), that is, prove that, ||h|| <= ||k||/\epsilon. This can be found on the last line of page 222.
Lecture 8 (August 21, 2023)
1. Remark on the proof of Inverse Function Theorem: In the derivation of equation (52) we have only used that f is differentiable on E, f'(a) is invertible and that f' is continuous at a. The use of the fact that f is in C^1 (E) is used to prove the continuity of g' on V.
2. Theorem 9.25: Let f be in C^1(E), f:E--> R^n, where is an open set in R^n. Suppose f'(x) is invertible for each x in E, then f:E--> R^n is an open map. That is, f maps every open set in E to an open set in R^n. (Proof is similar to the proof of the Inv Fun Thm and is left as an exercise.)
3. The hypothesis made in Theorem 9.25 ensure that each point x in E has a neighbourhood in which f is one-to-one. That is, f is locally one-to-one. But f need not be one-to-one on E. This can be illustrated by the following example which we discussed in class.
Consider f: R^--> R^2 given by f(x,y) =(e^x cos y, e^x siny)
(a) What is the range of f?
(b) Show that the Jacobian of f is not zero for any x in R^2. Thus every point on R^2 has a neighbourhood in which f is one-to-one (by the Inv Fun Theorem). But f is not one-to-one on R^2 as f(x,y) = f(x, y+2 \pi).
(c) Put a=(0, \pi/3), b=f(a). Let g be a cont inverse of f defined in a nbd of b such that g(b)=a. Find the explicit expression for g. Compute f'(a), g'(b) and verify that g'(b) = [f'(a)]^{-1}.
(d) What are the images under f of the lines parallel to the coordinate axes.
4. Definition of a graph: Let X and Y be metric spaces. Let E \subset X. Let f :E--> Y be a function. Then the graph of f is the set {(x, f(x)) | x \in E} \subset E X Y \subset X X Y.
5. Consider f: R^2--> R defined as f(x,y)= x62+y^2 -1. Then f^{-1}(0) =S^1. Put p=(0,1) \in f^{-1}(0). Consider the open set U={y>0} in R^2 then U intersected with f^{-1}(0) ={(x,y) \in S^1 with y>0} is an open neighbourhood of p in S^1. Then, (x,y) \in U intersection f^{-1}(0) iff y =\sqrt{1-x^2}. That is, U intersected with f^{-1}(0) = graph (h) where h:(-1,1 --> R is defined as h(x) = \sqrt{1-x^2}. Check that here the partial derivative of f wrt to y evaluated at p is nonzero.
We also discussed what happens at q=(1,0).
6. Exercise: Repeat similar arguments for f: R^3-->R defined as f(x,y,z) = x^2 +y^2 +z^2 -1 and p=e_3 =(0,0,1).
7. Statement of Implicit Function Theorem: Let U be an open subset of R^{n+k}. Let f:U--> R^k be a C^1 map. Suppose p \in U is such that f(p)=0 and the set {D_{n+1}f(p), ...., D__{n+k}(p)} is linearly independent.Let p=(a,b) \in R^n X R^k. Then there exists open sets V_1 of a in R^n and V_2 of b in R^k and a C^1 map g: V_1---> V_2 such that f^{-1}(0) intersect (V_1 X V_2) = graph (g). That is, (x,y) \in (V_1 X V_2) and f(x,y)=0 iff y=g(x).
Lecture 9 (August 22, 2023)
Proved the Implicit Function Theorem by applying Inverse Function Theorem to a proper "thickening" of f.
As a consequence of the Implicit Function Theorem we see that f^{-1}(0) is locally homeomorphic to an open subset of R^n.
Stated and proved the linear version of the Implicit Function Theorem (Theorem 9.27 of [Rudin]).
Defined (a) a surface in R^3 and (b) a parametrisation (U,F) of a surface S around a point p on S.
Theorem: If S is a surface in R^3 and if (U,F) is a parametrisation of S then, F^{-1} : F(U)--> U is a C^1 map.
We started to prove this theorem as an application of the Inverse Function Theorem to a proper "thickening" of f. We have invoked the InvFunThm in the class. Try to complete the proof on your own.
Lecture 10 (August 24, 2023)
1. Theorem: The local homeomorphism in the definition of a surface is in fact a diffeomorphism.
Completed the proof of this result as an application of inverse function theorem.
2. Applied the linear version of the Implicit Function Theorem to the projection map \Pi_n :U^{n+k} to R^n and proved that \Pi_n ^{-1} (0) = {(0,k) | k \in R^k} = graph of 0 function. Here. 0 is the zero vector of R^n, Note that by Open mapping theorem, \Pi_n is an open map. \Pi_n is surjective but not injective.
3. Let f: U^(n) ---> R^{n+k} given by f(x_1, .. x_n) =(y_1, ... y_n, y_{n+1},.... y_{n+k}) be a C^1 map. If the rank of f'(p) =n. If we define \tilde{f} :U^{n+k} ---> R^{n+k} as \tilde{f} (x_1, ... x_n, x_{n+1},.... x_{n+k} )= f (x_1, ... x_n) = (y_1, ... y_n, y_{n+1},.... y_{n+k} ) then we see that \tilde{f}'(p) is not invertible. Hence Inv Func Theorem isn't applicable.
4. We apply the inverse function theorem to \hat{f}:= \Pi_n composed with f: U^(n) ---> R^n. If f'(p) has a n X n submatrix A=a_ij= (dy_i/dx_j) (1<= i<= n, 1<= j<= n) which is invertible then we see that \hat{f}'(p) = A. Hence, Inv Func Thm applies to \hat{f} tells that there exists open nbds U1 of p in U^(n) and V1 of \hat{f}(p) in R^n with V1= \hat{f}(U1) such that \hat{f} :U1 --> V1 is a C^1 diffeomorphism.
Lecture 11 (August 28, 2023)
* Stated The (Constant) Rank Theorem: U is an open subset of R^m and V is an open subset of R^n. Let r<=m, r<=n. f : U --> V is a C^1 map such that the rank of [f'(x)]=r for each x \in U, that is, f' has a constant rank on U. Then, for each p in U there is a chart (U_0, \phi) centered at p (that is, \phi(p)=0) and a chart (V_0, \psi ) centered at f(p) with U_0 \subset U and f(U_0) \subset V_0 \subset V such that \psi composed with f composed with \hi^{-1} is \Pi_r, the projection in the first r coordinates. Proof is left as an optional exercise.
* Defined Derivatives of order 2 for a real valued function of several variables, defined real valued functions of class C'' on E, Defined vector values functions of class C'',
* Recalled Theorem 5.10: The Mean Value Theorem (MVT) for f:[a,b] --> R. Stated and proved the MVT (Theorem 9.40) for real valued function of two variables. Saw tha analogy between these two theorems.
* Stated and proved Theorem 9.41 which gives a situation when D_{12}f = D_{21}f at a given point of E.
* As a corollary to Theorem 9.41 it follows that for functions of class C'', the conclusion of Theorem 9.41 holds.
* Defined functions of class C^k on E inductively. Exercise: As a repeated application of Theorem 9.41, prove that for f \in C^k(E), D_{i_1, i_2, \ldots, i_k} f : = D_{i_1} D_{i_2} \ldots \ldots D_{i_k} f is unchanged if the subscripts i_1, i_2, \ldots, i_k are permuted. (Exercise 29).
Lecture 12 (August 29, 2023)
Sequence and Series of functions:
We will consider functions from a set E to IF, where F= IR or IC.
1. Definition of pointwise convergence of a sequence of functions
2. Definition of pointwise convergence of series of functions
3. Main concern: Whether important properties like continuity, differentiability or integrability are preserved under the limit operations.
4. Definition of uniform convergence of a sequence of functions on E, definition of uniform convergence of a series of functions on E, Difference between pointwise convergence and uniform convergence.
5. Theorem 7.8 (Cauchy's Criterion for uniform convergence)
6. Theorem 7.9 Equivalent of uniform convergence of {f_n} and convergence of the sup_{x \in E} |f_n(x) -f(x)| to 0.
7. Theorem 7.10 (Weierstrass' M Test)
8. Stated Theorem 7.11 (Uniform convergence and Continuity).
Lecture 13 (August 31, 2023)
* Proved Theorem 7.11
* Theorem 7.12 is an immediate corollary of Theorem 7.11. The statement is: If {f_n} is a sequence of continuous functions on E converging uniformly on E to a function f then f is continuous on E.
* Justified using an example that the converse of Theorem 7.12 is not necessarily true. That is, if a sequence of continuous functions on E converge pointwise to a continuous function on E, then the convergence is not necessarily uniform. Example, f_n(x)= n^2 x (1-nx) if x \in [0, 1/n) and is zero for x\in [1/n, 1]. Then each f_n is cont on [0,1] and converges pointwise to the constant zero function on [0,1]. But the convergence is not uniform.
* Theorem 7.13 (Dini's Theorem) gives a situation when the converse holds. Proved this theorem in the class.
* Gave a pictorial illustration (using graphs of functions and the graph of the limit function) of when a sequence of functions converge pointwise and when it converges uniformly.
* Exercise: Please illustrate using an example that the compactness of K is crucial in the statement of Theorem 7.13.
* Discussed a few examples of sequence of functions which converges uniformly on an interval, and examples where the convergence is pointwise but not uniform.
* Exercise: Please notice that the sequence of functions, straight line segments in R^2 with slope 1/n and with y-intercept 1, that is, f_n(x) = 1+ x/n, x \in [0,1] (defined in the class today), satisfy the hypotheses of Dini's Theorem and hence their convergence to the constant 1 limit function is uniform.
* Introduced the metric space (C(X), d), where C(X) is the set of all complex valued continuous and bounded functions on the metric space X and d(f,g):= \sup_{x\in X} |f(x)-g(x)|. Recalled that C(X) is a vector space under pointwise addition and scalar multiplication of functions.Checked that d is a metric on C(X). And indicated a proof of the fact (Theorem 7.15) (C(x),d) is a complete metric space. Please try to write down the proof on your own.
* Remark: Convergence of a sequence in (C(X),d) is the same as uniform convergence of the sequence of functions (under consideration) on X.
Lecture 14 (September 4, 2023):
Theorem 7.16 Uniform Convergencce and Integration
Example 7.5 A sequence of differentiable function converging uniformly to a differentiable function but we can not exchange the limit and the differentiation operations.
Theorem 7.17 Uniform Convergence and Differentiation
Theorem 7.18 There exists a continuous function f:R --> R such that f is nowhere differentiable.
Lecture 15 (September 5, 2023)
Definitions of point wise bounded and uniformly bounded sequence of functions on a set E.
Theorem 7.23 Every point wise bounded sequence of complex valued functions on a countable set E has a subsequence that converges pointwise for every x \in E.
Remark 1: Even if we have a sequence of functions in C(K), K compact, there need not exist a subsequence which converges pointwise on K.
Example 7.20 illustrating remark 1.
Question? Does every convergent sequence have a uniformly convergent subsequence?
Remark 2: A sequence of bounded functions may converge without being uniformly bounded.
Example 7.6 illustrating Remark 2.
Exercise 1: Prove that the sequence in example 7.6 is not uniformly bounded on [0,1].
Exercise 2: Prove that if {f_n} is a sequence of bounded functions on E which converge uniformly on E then {f_n} is uniformly bounded.
Remark 3: Exercise 2 tells us that the sequence of functions in example 7.6 does not converge uniformly on [0,1].
Remark 4: Even if {f_n} is uniformly bounded on a compact set K which converges pointwise on K theer need not exist a convergent subsequence.
Example 7.21 illustrating Remark 4.
Definition 7.22 Equicontinuous family of functions on a set E, where E is a subset of a metric space (X,d).
Remark 5: It follows immediately from the definition of equicontinuous family of functions on E that If f belongs to a family of equicontinuous functions on E then f is uniformly continuous on E.
Discussed examples of equicontinuous family of functions.
The sequence of functions given in example 7.21 is not an equicontinuous family on [0,1].
Theorem 7.24 Let K be a compact metric space. Let f_n \in C(K) for all n in IN. If {f_n}
converges uniformly on K then {f_n} is equicontinuous on K.
Remark 6: The converse of Theorem 7.24 need not be true.
Exercise 3: Find an example illustrating Remark 6.
Lecture 16 (September 7, 2023)
Recalled the following Theorems, Remarks and Exercises from the previous lecture..
Theorem 7.23 Every point wise bounded sequence of complex valued functions on a countable set E has a subsequence that converges pointwise for every x \in E.
Remark 1: Even if we have a sequence of functions in C(K), K compact, there need not exist a subsequence which converges pointwise on K.
Question? Does every convergent sequence have a uniformly convergent subsequence?
Remark 2: A sequence of bounded functions may converge without being uniformly bounded.
Exercise 1: Prove that the sequence in example 7.6 is not uniformly bounded on [0,1].
Exercise 2:: Prove that if {f_n} is a sequence of bounded functions on E which converge uniformly on E then {f_n} is uniformly bounded.
Remark 3: Exercise 2 tells us that the sequence of functions in example 7.6 does not converge uniformly on [0,1].
Remark 4: Even if {f_n} is uniformly bounded on a compact set K which converges pointwise on K theer need not exist a convergent subsequence.
Remark 5: It follows immediately from the definition of equicontinuous family of functions on E that If f belongs to a family of equicontinuous functions on E then f is uniformly continuous on E.
Theorem 7.24 Let K be a compact metric space. Let f_n \in C(K) for all n in IN. If {f_n}
converges uniformly on K then {f_n} is equicontinuous on K.
Remark 6: The converse of Theorem 7.24 need not be true.
Exercise 3: Find an example illustrating Remark 6. Solved this exercise.
(a) Noncompact set: Discussed the example f_n:R--> R, defined as f_n(x) = x/n. Check that this is an equicontinuous family of functions on R which converges pointwise but does not converge uniformly on R.
(b) Compact set: Please find an eequicontinuous family of functions on a compact set which does not converge uniformly on K.
An Example: Discussed the example f_n: [0,1] --> R defined by f_n(x) =sin nx then we have seen that f_n can not have a pointwise convergent subsequence on [0,1], therefore, f_n can not have a unif convergent subsequence on [0,1] and hence {f_n} does not converge uniformly on [0,1]. Check that {f_n} is not an equicontinuous.family of functions on [0,1]. Hint: Find e>0 and a pair of points x, y in [0,1] for which no delta (as in the definition of equicontinuity) works This sequence of functions is not equicontinuous on. [0,1] and hence, in view of Theorem 7.24, does not converge uniformly on [0,1].
Stated the Ascoli-Arzela Theorem (Theorem 7.25) and proved part (a) of this theorem.
Quiz 3 was conducted.
Lecture 17 (September 11, 2023)
1. Completed the proof of Ascoli Arzela theorem (Theorem 7.25)
2. Stated the Stone- Weiestrass Theorem (Theorem 7.26) and its Corollary (7.27). Indicated the proof of the corollary.
3. Started proving the proof of the Stone-Weiestrass Theorem. Will complete the proof in the next class.
4. Exercise: Please check that if, in the statement of corollary 7.27, |x| is replaced by any other continuous function f on the interval [-a,a], can we find a sequence of polynomials {P_n} such that P_n(0)=0 for each n \in |N and {P_n} converges to f uniformly on [-a, a]? Justify your answer. If your answer is No then is there any other property that f is supposed to satisfy for this to be true?
Lecture 18 (September 12, 2023)
1. Completed the proof of Stone-Weiestrass Theorem.
2*. Stated a Generalisation of this Theorem to an algebra of functions. (Theorem 7.32).
3*. Stated a Generalisation for complex valued continuous functions on K. (Theorem 7.33).
4*. Saw that the Stone Weiestrass Theorem (Theorem 7.26) stated and proved in the class is a particular case of the generalisations mentioned above.
5. Discussed Exercise 20 from [Rudin].
6. Recalled Theorems 7.23 and 7.24 and discussed Exercise 16 in view of these two Theorems.
7. Discussed Exercise 15 from Rudin and given a hint that prove that f is a constant function on [0, \infty).
8. Also, discussed why the sequence {sin nx} can't be an equicontinuous sequence as a corollary of Exercise 15.
9*. Recalled the Heine Borel Theorem. Also, stated that If X is a NLS then the following are equivalent:
(a) Every closed and boundede subset of X is compact.
(b) The closed unit ball in X is compact.
(c) X is finite dimensional.
10. Recalled that C(K), where K is a compact subset of a metric space (with infinitely many points), is not finite dimensional.
11. Consider C(K) with the sup norm. We have seen that C(K) is a complete metric space with the sup metric. That is, C(K, || ||_\infty) is a Banach space.
12. Exercise 19 of [Rudin] says that "A subset S of C(K) is compact if and only if S is uniformly closed, pointwise bounded and equicontinuous."
13*. Remark: A Banach space can not have a denumerable (countably infinite) Hamel Basis. A need for a more general notion of a basis, namely the "Schauder basis" was floated.
14*. Application of Ascoli-Arzela Theorem: The Fundamental Theorem of ODE, that is, the Peano's Theorem, that is, the existence of a solution of an initial value problem (IVP) in ODE.
The points marked with * are not mandatory for you to study or understand. They are just extra remarks made for you to get a broader perspective.
Portion for the MidSem Exam (September 21, 2023):
Chapter 9 Functions of Several Variables:
Section 9.6 (Definition of Operator Norm) to Section 9.28 (Implicit Function Theorem)
Operator norm of a linear map from R^n to R^m, Differentiation on R^n, definition of total derivative, Uniqueness of Total Derivative, Chain Rule, Partial Derivatives, Dorectional Derivatives, Gradient of a function f at a point x, Mean Value Inequality for a differentiable map on a convex open set of R^n, Continuously Differentiable maps or C^1- maps, Contraction principle, Inverse function theorem, implicit function theorem.
Chapter 7 Sequences and Series of functions:
Section 7.1 (Pointwise convergence of a sequence of functions) to Section 7.26 (Stone-Weierstrass Theorem)
Pointwise and uniform convergence of sequences of functions, examples, non-examples, Uniform convergence of series of functions, Weierstrass M-test, uniform convergence and continuity, uniform convergence and integration,, uniform convergence and differentiation, Pointwise bounded sequence of functions, uniformly bounded sequence of functions, Equicontinuous family of functions on a subset E of a metric space (X,d), Arzela-Ascoli theorem, Stone-Weierstrass theorem.
Lecture 19 October 5 (Thursday)
Recalled the definition (Definition 3.21) of a Series from chapter 3 of [Rudin].
Also recalled Theorem 3.25 (Comparison Test), Theorem 3.33 (The Root Test), Theorem 3.34 (The Ratio Test).
Recalled Definition of Power Series (Definition 3.38), Recalled Theorem 3.39 about radius of convergence and disk of convergence, Discussed a few examples about radius of convergence and the behaviour of the series on the boundary of the disk of convergence, Recalled Theorem 3.43 about convergence of an Alternating Series, Recalled Theorem 3.44 about the convergence of a particular kind of power series on the unit sphere when the radius of convergence is 1.
Recalled Absolute Convergence (AC) of a series, Recalled Theorem 3.45 which says that AC implies convergence (C) as it is. Recalled Remark 3.46 that for a series of positive terms AC is same as C. Illustrated with an example that C need not imply AC, in general.
Remark*: It is straight forward to see that.every power series converges absolutely in the interior of its disk of convergence.
Chapter 8 Some Special Functions
Defined an analytic function on |R,
Stated and proved Theorem 8.1 about the uniform convergence on [R-e, R+e] (for every e>0), continuity and differentiability in (-R, R) of an analytic function having radius of convergence R. And the fact that, in its interval of convergence, the derivative of an analytic function is the same as the limit of its power series expansion differentiated term by term.
To prove Theorem 8.1, we used Remark* mentioned above, the Weierstrass' M Test (Theorem 7.10) and Theorem 7.17 (relating uniform convergence and differentiation of a sequence of functions),
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