I have a stack of lecture notes that I am currently going through to teach myself a little bit about Fourier Analysis. Now I struggle with the following Lemma, which is needed to talk about the Inverse Fourier Transform in $S(\mathbb{R})$: Let $T : S(\mathbb{R}) \to S(\mathbb{R})$ be a linear map that satisfies $T(xf) = […]

I am trying to understand a part of the following proof. Prove that $(1)$ and $(2)$ are equivalent: $(1)$ $\lim_{x \to c}f(x)=f(c)$ $(2)$ $f$ is continuous at $c$. I understood the proof of $(1) \Rightarrow (2)$. My question is about a part of the proof of $(2) \Rightarrow (1)$. Definition Definition of continuity I am […]

If we have a Lissajous curve given by the parametric equations $$x(t)=A\sin(\omega_x t)$$ $$y(t)=B\sin(\omega_y t + \delta),$$ how can we show that the curve is dense in the rectangle of sides $A,B$ if and only if $\omega_x$ and $\omega_y$ are incommensurate (i.e. their ratio is irrational)? What I tried: Suppose it is not dense, so […]

Let $X$ be a infinite dimensional Banach space. How to construct a example of a continuous function $f:X\rightarrow\mathbb{R}$ such that $f$ is coercive, but is not bounded below. $f$ is coercive if $\|u\|\rightarrow \infty$ then $f(u)\rightarrow\infty$ $f$ is bounded below if there exist a constant $C$ such that $f(u)\geq C$ for all $u\in X$

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Possible Duplicate: Why is the number e so important in mathematics? Intuitive Understanding of the constant “e” The number $e$ is important in many respects. If you ask anyone why it is important, you will get multiple answers. Some say the defining property is that the derivative of $e^x$ is $e^x$. Some say that it […]

If $f \in \mathscr{R}\left(\alpha_1\right)$ and $f \in \mathscr{R}\left(\alpha_2\right)$, then $f \in \mathscr{R}\left(\alpha_1 + \alpha_2 \right)$ and $$ \int_a^b f d\left(\alpha_1 + \alpha_2 \right) = \int_a^b f d\alpha_1 + \int_a^b f d\alpha_2.$$ This is part of Theorem 6.12 (e) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Here is my proof: Let […]

assume {$a_n$}$_{n=1}^\infty$ ,$a_n$ is none negative and real sequence that satisfied :$$1+a_{m+n}\leq (1+a_{m})(1+a_{n}) ,\quad m,n\in\mathbb N$$ how prove $x_n=(1+a_{n})^\frac1n $ is convergent? thanks in advance

Lately I’ve been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? I’m having a tough time finding this online. For example, in $4$D, could we just take rotations in the planes $XY$, $XZ$, $YZ$, $XW$, $YW$, […]

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. Does anyone have an explicit formula for the inverse function $f^{-1}(\theta)=?, -\pi<\theta<0$? It seems unlikely, but I cannot be sure.

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