Intereting Posts

Combinatorial proof that binomial coefficients are given by alternating sums of squares?
Probability of throwing balls into bins
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
Sentence such that the universe of a structure has exactly two members
Why is one “$\infty$” number enough for complex numbers?
Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$
$X$ Poisson distribution, $Y$ geometric distribution – how to find $P(Y>X)$?
Strength of the statement “$\mathbb R$ has a Hamel basis over $\mathbb Q$”
Optimal distribution of points over the surface of a sphere
How a group represents the passage of time?
Class Group of Ring of Integers of $\mathbb{Q}$
(co)reflector to the forgetful functor $U:\mathbf{CMon} \to \mathbf{ Mon}$
Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
Converse of interchanging order for derivatives
If $f(a) = g(a)$ and $f'(x) < g'(x)$ for all $x \in (a,b)$, then $f(b) < g(b)$

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm

they are not locally convex, since the only convex neighborhood of zero is the whole space

Why is this so? http://en.wikipedia.org/wiki/Lp_space

- Distinguishing between symmetric, Hermitian and self-adjoint operators
- At most finitely many (Hamel) coordinate functionals are continuous - different proof
- Image of unit ball dense under continuous map between banach spaces
- Compact Metric Spaces and Separability of $C(X,\mathbb{R})$
- TVS: Uniform Structure
- The group of invertible linear operators on a Banach space
- Direct sum of orthogonal subspaces
- Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$
- Existence of smooth function $f(x)$ satisfying partial summation
- Showing Lipschitz continuity of Sobolev function

**Key fact.** Given a function $f\in L^p([0,1])$ and a positive $\epsilon>0$, one can write $f$ as a finite convex combination of functions $g_1,\dots,g_n$ such that $\|g_k\|_{L^p}<\epsilon$ for all $k=1,\dots, n$.

Having the above, we conclude as follows: if $U$ is a convex neighborhood of $0$, then it contains the set $\{g: \|g\|_{L^p}<\epsilon\}$ for some $\epsilon>0$. The above fact, together with convexity, imply that $U$ contains all $L^p$ functions.

Proof of the key fact (sketch): For any $n$ there is a partition of $[0,1]$ into intervals $J_1,\dots,J_n$ such that $\int_{J_k} |f|^p=n^{-1}\int_0^1|f|^p$. Let $g_k=n\,f\,\chi_{J_k}$. Calculate $\|g_k\|_{L^p}$ and observe that it tends to $0$ as $n\to\infty$.

- To find all odd integers $n>1$ such that $2n \choose r$ , where $1 \le r \le n$ , is odd only for $r=2$
- Xmas Combinatorics 2014
- Why is this proof of the chain rule incorrect?
- $\ker T\subset \ker S\Rightarrow S=rT$ when $S$ and $T$ are linear functionals
- Vertical bar sign in Discrete mathematics
- Neumann problem for Laplace equation on Balls by using Green function
- Prove that every undirected finite graph with vertex degree of at least 2 has a cycle
- $G$ be a non-measurable subgroup of $(\mathbb R,+)$ ; $I$ be a bounded interval , then $m^*(G \cap I)=m^*(I)$?
- Tensor Calculus
- Unit circle is divided into $n$ equal pieces, what is the least value of the perimeters of the $n$ parts?
- Graph of symmetric linear map is closed
- How to prove Poincaré-like inequality for the integral over the boundary?
- Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?
- diffusion equation, inhomogenous boundary conditions (the subtraction method)
- Prove the statement 'World is not flat'