Articles of functional analysis

Is $GL(E)$ dense in $L(E)$, when $\dim E=\infty$?

Let $E$ be a normed vector space (Banach space, if you like). Is $GL(E)$, the set of invertible and continuous endomorphism of $E$, dense in $L(E)$, the set of continuous endomorphism of $E$? I specify that I know the answer if $dim(E)<\infty$, with classical arguments about the spectrum of matrices, and, I know that $GL(E)$ […]

Normal derivative of a $H^1$- Sobolev function

Let $u\in H^1(\Omega)$, where $\Omega$ is a bounded open set of $\mathbb{R}^n$ with Lipschitz boundary. We denote the outward unit normal as $n$, defined a.e. on $\partial\Omega$, and the normal derivative of $u$ as $$ \frac{\partial u}{\partial n}:=\nabla u\cdot n. $$ Which space does the normal derivative belong to? Is it possible to show $\frac{\partial […]

Functions by which one can multiply elements of $L^1_{\text{loc}}$

Let $u$, $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We further assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ for any $x\in\mathbb R^N$ and $r>0$. My question 1: do we have $u\omega\in L^1_{\text{loc}}(\mathbb R^N)$ as well? I feel not but can not find an counterexample…

Coordinate functions of Schauder basis

If $X$ is a Banach space admitting a Schauder basis, then can we choose a set $\{e_1,e_2,e_3 \cdots \}$ as the basis such that there are bounded linear functional $f_i$ such that $f_i(e_j)=\delta_{ij}$?

Inner product is jointly continuous

I’m attempting another exercise from my notes: Show that an inner product on an inner product space is jointly continuous with respect to the induced norm:if $v_n \to v$ and $w_n \to w$ as $n \to \infty$, then $\langle v_n, w_n\rangle \to \langle v,w \rangle$ as $n \to \infty$. I’d not heard jointly continuous before […]

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with the components) Let $r=l+\alpha$ with $l \in \mathbb{N}$ and $0\le \alpha <1$. Case 1) If $\alpha >0$, $C^r_{*}(\Omega)$ is […]

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don’t remember encountering such a question in calculus or ODE (at least not the most important problems) Further more all […]

Fixed point in a continuous map

Possible Duplicate: Periodic orbits Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$. Does $f$ necessarily have a fixed point?

polynomial approximation in Hardy space $H^\infty$

$H^\infty$ is the Hardy space of bounded analytic functions on the open unit disk $|z| < 1$ with the norm $$\|f\| = \sup_{ |z| \,< \,1}|f(z)|$$ It has two important subspaces : $H^\infty_C$ the (closed) subspace of analytic functions that stay continuous on the closed unit disk $|z|\le 1$. $H^\infty_K $ the subspace of functions […]

separation theorem for probability measures

Suppose I have a probability measure $\nu$ and a set of probability measures $S$ (all defined on the same $\sigma$-algebra). Are the following two statements equivalent? (1) $\nu$ is not a mixture of the elements of $S$. (2) There is a random variable $X$ such that the expectation of $X$ under $\nu$ is less than […]