# Tagged Questions

5 views

38 views

### Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
27 views

### Approximating a $C^1$ function by piecewise affine maps

Let $\Omega\subset\mathbb R^n$ be an open and bounded domain and let $f\in C^1(\bar\Omega,\mathbb R^m)$. I would like to approximate $f$ by a function $u:\bar\Omega\to\mathbb R^m$ that is piecewise ...
33 views

### $f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
45 views

### Can I apply Holder's inequality in this case?

I would like to prove that for any $0<q \leq p$, if $x_1 \geq x_2 \geq ... \geq x_n \geq 0$ then \left( \sum_{j=m+1}^n x_j^p \right)^{1/p} \leq m^{\frac{1}{p} - \frac{1}{q}}\left( \sum_{j=1}^n ...
I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...