3
votes
2answers
78 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
0
votes
1answer
33 views

The Landau symbol $\mathcal{o}$ as in Königsberger Analysis I

I am currently working on Chapter 14 - local approximations of function and Taylor polynomials - in Königsberger Analysis 1 Background: Königsberger introduced the Taylor Polynomial of order ...
4
votes
1answer
71 views

Riemann Sums as in Königsberger Analysis 1

Intro: I must take a small detour here which is only relevant if you do not know the book itself and care about my background. I am working with Königsberger Analysis I (can be found on Springerlink). ...
1
vote
0answers
49 views

Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus ...
2
votes
1answer
24 views

Minimum distance from origin in $\mathbb{R}^3$ but Hessian is indefinite

Problem: Consider the Set $M= \lbrace (x,y,z) \in \mathbb{R}^3 \mid x^2+2y^2-z^2=1 \rbrace$ and find all Points on $M$ which have minimal euclidian distance from the origin. My approach: I ...
4
votes
4answers
160 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
8
votes
1answer
173 views

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
1
vote
2answers
65 views

Approaches to teaching and learning analysis

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take. IMHO, the ...
1
vote
1answer
30 views

Derivative of Normal Vector Field

This is an example from Do Carmo (Example 4, page 139). Consider the saddle point $p=(0, 0, 0)$ of the hyperbolic paraboloid $z=y^2-x^2$ with parameterization $\mathbf x(u, v)=(u, v, v^2-u^2)$. It is ...
2
votes
1answer
63 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
3
votes
2answers
139 views

Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
1
vote
2answers
47 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
0
votes
1answer
31 views

Is norm of a differentiable function continuous?

The following argument is from my class notes. "Suppose $\gamma(s): I\rightarrow \mathbb R^3$ is a regular curve. Fix $t_0$ and define $s(t) = \int_{t_0}^t \|\gamma'(t)\|dt$. That is, $s(t)$ is the ...
1
vote
1answer
19 views

Manipulation of $C^1$ funcitons

I think I read somewhere that composition of $C^1$ functions is also $C^1$, but I could not find the reference now. Also, is the difference of two $C^1$ functions still a $C^1$ function, please? And ...
0
votes
3answers
77 views

Counter Example about Continuous Functions

(James Munkres page 104 Theorem 18.1) Let $X$ and $Y$ be topological spaces; let $f: X \rightarrow Y$. If $f$ is continuous, then for every subset $A$ of $X$, one has $f(\overline{A})\subset ...
1
vote
1answer
54 views

How is an open set defined without referring to any distance function in a topology?

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly ...
2
votes
0answers
118 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
0
votes
1answer
354 views

Brouwer's Fixed Point theorem proof for 2-dimension

I am trying to find a elementary proof of the Brouwer's fixed point theorem only using basics of point set topology and real analysis. In the one of the textbooks I read, they were proving Brouwer's ...
2
votes
0answers
32 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
1
vote
1answer
57 views

show $\int g\log (g/f)$ is $0$ only if $g=f$ almost everywhere

Question: Suppose that $f$ and $g$ are two probability density functions, show that $\int g\log (g/f)$ is always non-negative and equals to $0$ $\it only\ if$ $\ g=f$ almost everywhere. I have ...
2
votes
3answers
98 views

Gamma function can be extended as a meromorphic function

Let $$\Gamma (z)= \int_{0}^{\infty} e^{-t}t^{z-1}dz$$ for $\Re z\gt 0$ Can be extended as a meromorphic function to the entire complex plane with simple poles at non positive integers. How can I ...
0
votes
1answer
26 views

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$ I know the result. But I dont know how to show this step by step.
7
votes
1answer
94 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
1
vote
1answer
52 views

Zeta function in complex analysis.

Show that $$\frac{\zeta'(z)}{\zeta(z)}=-\sum_{n=2}^{\infty}\frac{f(z)}{n^z}$$ for $\Re z\gt 1$ Where $f(z)= \ln p$ if $n=p^m$ for some prime $p$ and some $m\in \Bbb N^+$ Or $f(z)=0$ otherwise. ...
1
vote
2answers
84 views

Infinite product convergence

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges. I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ ...
2
votes
1answer
98 views

Prove that $\Gamma'(1)=-\gamma$

Use the product formula for $1/\Gamma(z)$ to prove that $$\Gamma'(1)=-\gamma$$ I know that for Euler constant $\gamma$, $$\frac{1}{\Gamma(z)} =ze^{\gamma z}\prod _{k=1}^{\infty} ...
1
vote
4answers
168 views

The complex gamma function

Show that $$\Gamma (z+1)=z\Gamma (z)$$ $\forall z\in \Bbb C$ except for $z=-n$ where $n\in \Bbb N$. I know that the gamma function is defined as $\Gamma (z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$ And ...
1
vote
2answers
145 views

Gamma function in complex analysis.

Prove that $$ \Gamma\left(z\right) = \lim_{n\to \infty}\int_{0}^{n}t^{z - 1}\left(1 - {t \over n}\right)^{n}\,{\rm d}t \quad\mbox{for}\quad \Re z \gt 0 $$ I know that $$ {\rm e}^{-t/n} = 1 - {t ...
3
votes
1answer
160 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...
1
vote
1answer
143 views

$f(z)$ has infinitely many zeros and that each zero is simple.

Let $f(z)=e^z-z$ I want to check $f(z)$ is finite order. And how to show that $f(z)$ has infinitely many zeros and that each zero is simple. Dfn: an entire function f is finite order if ...
1
vote
0answers
40 views

Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
4
votes
1answer
147 views

Is it possible to learn differential topology before analysis?

Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will ...
8
votes
0answers
265 views

How much time is reasonable to complete baby Rudin?

I've been teaching myself math for more than a year. My current aim is towards algebraic topology and differential geometry. Apart from a messy (by which i mean some rigorous and some not) ...
0
votes
1answer
45 views

To alternatively prove the theorem(*) by proving that $g^{(n+1)}(z_0)=0$ $\forall z_0\in \Bbb C$

Assume that $g=x+iy$ be an entire function. By a theorem(*), $\vert x(z)\vert \le N \vert z\vert ^n \ \ \forall z$ large enough and for constant $N\gt 0$ and for non-negative $n\in \Bbb Z$ ...
0
votes
2answers
79 views

Cauchy integral formula in complex analysis

Assume $g$ be an entire function. And $\exists \ n\gt 0 \:and\ n\in \Bbb Z $ and also $\exists N \: and\ M \in \Bbb R$ s.t. $\forall z \in \Bbb C , \ \ \vert z\vert \ge M\ \ \: and\ \ \ \vert ...
2
votes
2answers
77 views

The conditions for Harmonics functions in complex analysis

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$. What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$. And I want to ...
3
votes
1answer
142 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...
7
votes
3answers
2k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
0
votes
1answer
166 views

Contraction Mapping, why constant and weak inequality?

From Wikipedia, a contraction mapping is a function $f: M \rightarrow M$ on a metric space $(M,d)$ such that there exists a nonnegative real number $k<1$ such that for all $x,y\in M$, $$ d ...
2
votes
1answer
58 views

Prove or disprove a set $F$ is closed.

This is an example in my book that talks about $F$ being precompact; Let $F$ be the subset of $C([0,1])$ that consists of functions $f$ of the form $$f(x) = \sum_{n=1}^{\infty}a_n\sin(n\pi x) ...
1
vote
2answers
37 views

Let $Tx = 1+\log(1+e^x)$. Show that $T$ has no fixed points.

Let $Tx = 1+\log(1+e^x)$. Show that $T$ has no fixed points. This is what I have: We say that $T$ has a fixed point if $Tx=x$. $$Tx = 1+\log(1+e^x) = x$$ $$\log(1+e^x) = x-1$$ $$1+e^x = e^{x-1}$$ ...
1
vote
3answers
43 views

Prove that $\nabla\langle Ax,Ax\rangle = 2A^TAx$

Prove that $\nabla\langle Ax,Ax\rangle = 2A^TAx$. My book uses this property to prove the $2-norm$ of a matrix $A$ is the square root of the spectral radius of $A^TA$. That is $$||A||_2 = ...
0
votes
1answer
68 views

Let $\delta$ be a linear functional equipped with the sup-norm. Show that $\delta$ is bounded and compute its norm.

Let $\delta:C([0,1])\rightarrow\mathbb{R}$ be the linear functional at the origin: $\delta(f) = f(0)$. If $C([0,1])$ is equipped with the sup-norm $$\|f\|_{\infty} = \sup_{0\leq x\leq 1}|f(x)|.$$ Show ...
1
vote
1answer
32 views

What is the Density Theorem in this context?

I have this exercise: Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and ...
2
votes
1answer
32 views

Prove that $(|u-s|+|x-y|)^2\leq 2|u-s|^2+2|x-y|^2$.

Prove that $(|u-s|+|x-y|)^2\leq 2|x-y|^2+2|u-s|^2$. My professor used this inequality for a proof last week. How would one prove this? I thought about using the Cauchy-Swartz inequality. This is ...
2
votes
1answer
59 views

Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$.

I want to Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$. Where $A$ is a matrix. I have a proof that involves Jordan Blocks. The proof is long and involved but it not ...
2
votes
2answers
52 views

Proving that an operator $K$ is bounded and $||K|| = \max_{0\leq x\leq 1}\bigg\{\int_0^1|k(x,y)|dy\bigg\}$

Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and $$||K|| = \max_{0\leq ...
0
votes
1answer
40 views

Showing that a set $D$ is closed and open

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) \mid |t-t_0|\leq T, |u-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
2
votes
1answer
55 views

Help with understanding a proof on Ordinary Differential Equations

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) | |t-t_0|\leq T, |u(t)-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
0
votes
3answers
84 views

Question on Proof of the Contraction Mapping Theorem

Contraction Mapping Theorem If $T\colon X\to X$ is a contraction mapping on a complete metric space $(X,d)$ then there is exactly one solution $x\in X$. Proof: Let $x_0$ be any point in $X$. We ...