Tagged Questions

4
votes
2answers
52 views

Intuition behind the difference between derived sets and closed sets?

I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
4
votes
1answer
81 views

Discreteness of eigenvalues for certain operators - can this approach be made rigorous?

I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
0
votes
2answers
59 views

Examples of convergence of series

These questions are practice questions from the text "Advanced Calculus, Folland" chapter 6.2 (not HW) I am working on some exercises on convergence of series and I feel that I understand it well but ...
1
vote
1answer
148 views

Riemann–Stieltjes Integral

Can you give me a geometric intuition for the Riemann–Stieltjes Integral? Just like the Riemann Integral is an approximation for an area under a curve.
2
votes
3answers
121 views

Limit Question - Explanation

The limit of $f(x) = x$, as $x$ tends to zero is zero. What's the limit of the function $\dfrac{x^2}{x}$ as $x$ tends to zero? and What's the limit of the function f(x) = (modulus of x)/x ? I am ...
1
vote
1answer
118 views

What is a product $\sigma$-algebra?

My question is relatively simple: what is a product $\sigma$-algebra? And why they are important? Can anyone suggest any links of intuitive (possibly with simple figures) explanations? Or, maybe ...
6
votes
0answers
184 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
1
vote
2answers
71 views

visualisation of pointwise boundedness

A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$ ...
0
votes
1answer
157 views

Explaining this particular version of the Implicit Function Theorem

I understand the general 'word' definitions of the Implicit Function Theorem and the simple examples such as the one on wikipedia but the version of the Implicit Function theorem given in our lecture ...
9
votes
1answer
306 views

Information captured by differential forms

My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
1
vote
3answers
225 views

Choice of $\xi$ [duplicate]

Possible Duplicate: Rational Numbers Suppose $\{x \in \mathbb{Q}|x>0,x^2<2\}$ has a supremum. Call this supremum $c$. In order to show that this cannot be the case, we learned that we ...
5
votes
0answers
153 views

Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
10
votes
4answers
2k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
11
votes
3answers
271 views

Nasty examples for different classes of functions

Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Usually when proving a theorem where $f$ is assumed to be continuous, differentiable, $C^1$ or smooth, it is enough to draw intuition by assuming ...
0
votes
0answers
142 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
0
votes
3answers
717 views

What is the norm measuring in function spaces

In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors. The generalization to function spaces is quite a mental leap (at ...
9
votes
3answers
856 views

Intuition for uniform continuity of a function on $\mathbb{R}$

I understand the formal definition of uniform continuity of a function, and how it is different from standard continuity. My question is: Is there an intuitive way to classify a function on ...
5
votes
3answers
544 views

Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
5
votes
1answer
336 views

Harmonic mean and logarithmic mean

The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $$H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + ...
10
votes
4answers
2k views

Meaning of convolution?

I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give ...
2
votes
2answers
356 views

Understanding of convergence of intersections of sets

If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of (-1/n, 1/n) as n->infinity)) However, at no time during the sequence is ...
8
votes
2answers
697 views

Example: Function sequence uniformly converges, its derivatives don't

Could anyone give an example of a sequence of differentiable (real) functions that uniformly converges to a differentiable function, but the derivatives of which don't converge to the derivative of ...
1
vote
1answer
267 views

Is this interpretation of Stieltjes integration correct?

If $f$ is a positive function, the intuitive interpretation of the Riemann integral $\int_a^b f(x) dx$ is the area under the curve $f$ between $a$ and $b$. Suppose $f$ and $g$ are smooth positive ...