Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking for a nice proof of the vector-valued version of Cauchy-Schwarz, $$ \langle f,g \rangle_2^2 = \left( \int f^{T}(t) g(t) dt \right)^2 \leq \langle f,f […]

I still want to examine different convergence results of the sequence of functions $f_n(x)= e^{-\frac{x^2}{n^2}}$ on $\mathbb{R}$. Convergence in the sense of distributions is shown in the following post: Convergence in $\mathcal{D}'(\mathbb{R})$ . Now I want to check the convergence in Convergence in $L^p(\mathbb{R})$ for $1\leq p \leq \infty$. How would you start the analysis? […]

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} \int_{\mathbb R} |f(x/r)| dx = r^{-1}\int_{\mathbb R} |f(y)| r dy= \int_{\mathbb R} |f(x)| dx;$ and, $\hat{f_{r}} (\xi)= \int_{\mathbb R} f_{r}(x) e^{-2\pi i \xi \cdot x} dx= \hat{f}(r\xi); (\xi […]

I’m trying prove this proposition, i got a part. Proposition: Let $(X,M, \mu)$ a finite measure space and define a metric space $M’$ as follows: for $A,B \in M$, define: $d(A,B) = \mu(A\triangle B)$. The space $M’$ is defined as all sets in $M$ where sets $A$ and $B$ are identified if $\mu(A\triangle B) = […]

Problem statement: Suppose $\mu$ is a finite measure on the Borel sets of $R$ such that $f(x) = \int_R f(x + t) \mu(dt)$ a.e., whenever $f$ is real-valued, bounded, and integrable. Show $\mu(\{0\}) = 1$ . My attempt at a solution: So, to begin with, I don’t fully understand the problem statement. Is $\mu(R) < […]

The function $f$ on $[0,1]$ is absolutely continuous on $[\epsilon,1]$ for $0<\epsilon<1.$ I further have that $$\int_0^1x|f'(x)|^pdx<\infty.$$ I’m trying to show that $$ \lim_{x\to 0}f(x)\ \text{exists and is finite}\qquad \text{if}\ p>2, $$ $$ \frac{f(x)}{|\log x|^{1/2}}\to 0\ \text{as}\ x\to 0\qquad \text{if}\ p=2,\ \text{and} $$ $$ \frac{f(x)}{x^{1-\frac{2}{p}}}\to 0\ \text{as}\ x\to 0\qquad \text{if}\ p<2. $$ (Of course, for […]

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in L_1\iff\sum_{n=1}^{\infty}n\mu(E_n)<+\infty$$ More generally for $1 \le p <\infty$ show that: $$f\in L_p\iff\sum_{n=1}^{\infty}n^p\mu(E_n)<+\infty$$ Here is what I got by reading an intempt to prove the statement: $$\chi_{E_1} + \frac{1}{2} \sum_{n=2}^{\infty}n\chi_{E_n}\le \sum_{n=1}^{\infty}(n-1)\chi_{E_n}\le|f|\le […]

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every $j\geq 1$. I’ve proved $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ easily. But for converse I stuck. please help me. I can not put any comment. so I have to write here. Thanks to triple_sec.

I’m learning about Fourier analysis and need help with the following problem (which is part of a subchapter on $L^p$ spaces): Using the Cauchy-Schwarz inequality show that if $f \in L^2[-\pi, \pi]$, then $(1)$ $f \in L^1[-\pi, \pi]$ and $(2)$ $\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$. My work and thoughts: $(1)$ We note that if $f \in […]

I’m searching for an example for $u\in W^{1,1}(\Omega)$, $f\in C^1(\mathbb{R})$ such that the composition $f\circ u\notin W^{1,1}(\Omega)$, where $\Omega\subset \mathbb{R}^n$ is open, bounded. I know that if $f\in C^1(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$, then it’s not possible to find such an example. And I know that $\frac{1}{x}$ is not in $W^{1,1}(0,1)$ for example, so that I tried to […]

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