Let $u : \mathbb{R} \to [-1, 1]$ be defined by $u(y) = \cos(2\pi y)$.
Set $u_\epsilon(x) = u\left(\frac{x}{\epsilon}\right)$ for $x \in (a,
> b)$, where $a, b \in \mathbb{R}$ and $a < b$. Show that $ u_\epsilon \rightharpoonup 0 $ in $ L^2(a, b) $ as $ \epsilon \to 0 $.
Solution
With $u(y)$, ...
I want to prove that the following inequality holds in an arbitrary metric space:
$$\overline{uz}\cdot\overline{xz} + \overline{uz}\cdot\overline{yz} \leq \overline{xz}\cdot\overline{ux} + \overline{yz}\cdot\overline{uy} $$
Where $\overline{st}$ is shorthand for $d(s,t)$. The only assumption abo...
Let $u : \mathbb{R} \to [-1, 1]$ be defined by $u(y) = \cos(2\pi y)$.
Set $u_\epsilon(x) = u\left(\frac{x}{\epsilon}\right)$ for $x \in (a,
> b)$, where $a, b \in \mathbb{R}$ and $a < b$. Show that $ u_\epsilon \rightharpoonup 0 $ in $ L^2(a, b) $ as $ \epsilon \to 0 $.
Solution
With $u(y)$, ...
