Articles of analysis

Tensor product of Hilbert Spaces

I am following this link under “definitions” I need to see why the suggested inner product on the pre-Hilbert space $H_1$ tensor $H_2$ is well defined. Recall that the fundamental tensors are a spanning set, but not a linearly independent one, hence not free in the category of vector spaces. How do I know that […]

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this statement is true, so I tried to the following proof. For any […]

Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions. My attempt: I don’t know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$. Can anyone help?

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum’s outline to this solution and it defines: $f(z) = \displaystyle \frac{1}{z^6 +1}$ And it says consider a closed contour $C$ Consisting of the line from $-R$ […]

Optimal assumptions for a theorem of differentiation under the integral sign

Let us consider the following integral: $$I(x)=\int_\Omega f(x, \omega)\, d\omega, $$ where $\Omega$ is a measure space and $f\colon \mathbb{R}\times \Omega \to \mathbb{R}$ is such that $f(x, \cdot)\in L^1(\Omega)$ for all $x$. When can we differentiate $I$? A dominated convergence argument gives the following result. Proposition. If For almost all $\omega\in \mathbb{\Omega}$, $$f(\cdot, \omega)\ \text{is […]

$\lim_{x \to 0} \dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} = 1$ for any $f,g \in C^1$ that are tangent to $\text{id}$ at $0$ with some simple condition

Theorem For any real functions $f,g \in C^1$ such that $f(0) = g(0) = 0$ and $f'(0) = g'(0) = 1$ and $x$ is strictly between $f(x)$ and $g(x)$ for any $x \ne 0$:   $f,g$ are invertible on some open neighbourhood of $0$   $\dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} \to 1$ as $x \to 0$ Questions What is […]

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family $\{E_{i}\}$ of pairwise disjoint Borel sets of $\mathbb R$), and for which $\mu(E)$ is finite if the closure of $E$ is […]

$1/r^2\int_{\mathbb{S}_r}u-u(x)$ converging to $\Delta u(x)$?a

When reading some papers on PDEs, the following shows up several times: For a $C^{\infty}$ function $u$, $\frac{\int_{\mathbb{S}_r(x)}u-u(x)}{r^2}$ converges to $1/2n\Delta u(x)$ uniformly on compact sets as $r\to 0$. Here $\mathbb{S}_r(x)$ is the sphere of radius $r$ centred at $x$, and $n$ is the dimension of the Euclidean space. I am wondering how we can […]