Issue with the definition of derivative in normed vector spaces
Let $(V,\Vert\cdot\Vert_V)$ and $(W,\Vert\cdot\Vert_W)$ be normed vector spaces and let $f\colon U\to W$ be a function defined on an open set $U\subseteq V$. $f$ is said to be differentiable at $x\in...
View ArticleProve that $\limsup a_n b_n = a \limsup b_n$ given $a_n > 0$ and $\lim_{n \to...
Let $a_n$ and $b_n$ be two sequences of real numbers. Assume that $a_n > 0$ and$\lim_{n \to \infty} a_n= a > 0$. Prove that $\limsup a_n b_n = a \limsup b_n$.Actually, I know what to do if...
View ArticleConvolution of non-differentiable function with uniform density
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous but non-differentiable function. If $g: \mathbb{R} \to \mathbb{R}$ is a function that is differentiable almost everywhere on $\mathbb{R}$, or if it is...
View ArticleEquivalence of Abel's inequality in integral form and sequence form
Abel Inequality (Sequence form) The sequence $a_n$ is monotonic then$$\left|\sum_{k=1}^n a_k b_k\right| \leq \max _{k=1, \ldots, n}\left|\sum_{i=1}^k...
View Articlecurve integral - intersection between plane and sphere
I am going to calculate the line integral$$ \int_\gamma z^4dx+x^2dy+y^8dz,$$where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation...
View ArticleRole of Second Derivative Continuity in Local Extrema, and Insights from...
Suppose that $x_0$ is stationary pointof a given function $\text{ i.e. } f'(x_0) =0$or in other words, it's differentiable at $x_0$ and$x_0$ is some point where $f$ has a local minimum /maximum,...
View ArticleIs h in the limit definition of the derivative a constant or a variable?...
Whenever someone explains the limit definition of the derivative they always talk about how $h$ approaches $0$.I get the impression that h is a variable.
View ArticleContinuouty of this functional series.
Let $U_{n}$ be the sequence of rational numbers and :$f(x) = \sum_{\left\{ n: U_{n} < x \right\}}^{} \frac{1}{2^{n}}$$\quad $ Show that $f$ is continuous on the irrationals, and it's not continuous...
View ArticleInverse Function theorem on closed sets
Let $U \subseteq \mathbb{R}^n$ open subset of $\mathbb{R}^n$ and $f:U\to\mathbb{R}^n$ be a $C^\infty$- function. Suppose for every $x\in U$ the derivative of at $x$, $df_x$ is non singular. Which of...
View ArticleDo these summations prove the limit properly?
I'm trying yo prove that if a sequence is convergent, then the Cesaro summantion of means is also convergent, I've gotten a decent proof outline I think is ok, but my inexperience with real analysis is...
View ArticleStrange Version of L'Hospital's Rule for Complex Functions
I have the task of adapting L'Hospital's rule to complex-functions $f(x)$ and $g(x)$, where $x \in \mathbb{R}$. The new theorem should look like:Suppose $f$ and $g$ are complex-valued functions and are...
View ArticleHow to show that the Gibbs measure is a stationary distribution of the...
I try to get the stationary distribution $$\pi=e^{-H(x)}dx/Z$$ and $Z=\int e^{-H(x)}dx$ called Gibbs measure for the following SDE:\begin{equation}\label{eq:ld} dX_t=-\nabla H(X_t)dt+\sqrt{2}dB_t, \,...
View ArticleNotation for Function Sequence Spaces
I wonder, is there any common notation about the function spaces.For numerical sequences we have some notations such as$c:= \{ (x_n) : (x_n)\,\ \text{convergent sequence} \}$$c_0:= \{ (x_n) : (x_n)...
View ArticleFind the $\lim_{x\to\infty}\frac{(1+1/x)^{x^2}}{e^x}$
Find the $$\lim_{x\to\infty}\frac{\left(1+\dfrac{1}{x}\right)^{x^2}}{e^x}.$$I know that the result is equal to $e^{-1/2}$, but I am curious why I can't substitute one of the $x$ outside result in...
View ArticleTranslation invariance of the Lebesgue measure
How to prove that the Lebesgue measure $m$ is translation-invariant for all Lebesgue measurable sets?I know that $m(B + x)=m(B)$, for Borel sets $B$ and $x \in \mathbb{R}^n$. So, I am trying to use the...
View ArticleClarifications involving topologist's sine curve
The typical topologist's sine curve is defined as,\begin{align}\label{1}\tag{1}& S = X \times Y = \{(0,y)| -1 \leq y \leq 1 \} \times \{(x, \sin (1/x))| x>0\}\\& X = \{(0,y)| -1 \leq y \leq...
View ArticleDivergence of product using tensor notation in the context of Linear Elasticity
I am trying to come up with a formula similar to the product rule for the divergence (see, for example, here) for the tensors appearing in the equations of Linear Elasticity in a manner similar to this...
View ArticleShow bijectivity of $f: (0,\infty)\to\mathbb{R},\quad x\mapsto\frac{x^2-1}{2x}$
How do I formally show that$$f: (0,\infty)\to\mathbb{R},\quad x\mapsto\frac{x^2-1}{2x}$$is bijective?Surjectivity:I solve $y=\frac{x^2-1}{2x}$ for $x$ so that $x\in (0,\infty)$. This is the case...
View ArticleUsual way of defining Reflection in Plane
We shall call a set $P(a,t)$ a plane in $\hat{\mathbb{R}^n}$(Extended plane) if it is of the form $P(a,t)=\{x\in \mathbb{R}^n: (x.a)=t\}\cup \{\infty\},$ where $a\in\mathbb{R}^n$,$a\ne 0,(x.a)$ usual...
View ArticleMonotone convergence theorem by Fatou's lemma
I want to prove the monotone convergence theorem using Fatou's lemma (and its reverse) as exercise, and I need a check; I will use also the following properties of limit inferior and limit superior:...
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