Articles of functional analysis

Dual space of a closed subspace of a Hilbert space

I’m reading Girault and Raviart’s book concerning Finite Element Methods for Navier-Stokes equations, and they use in the proof of one result, the following argument: As $V=\{v\in H_0^1(\Omega)^N; {\rm div}\;v=0\; \text{in}\;\Omega\}$ is closed in $H_0^1(\Omega)^N$, then V* (dual space of V) can be identified with a subspace of $H^{-1}(\Omega)^N$, where $\Omega$ is an open set […]

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f’$. Suppose $(f_n, f_n’) \to (g, h)$ in $X \times Y$. Thus, $f_n \to g, f_n’ \to h$ both uniformly on $[0, […]

Counterexample for “the sum of closed operators is closable”

I’m looking for a counterexample in a Banach space. I’ve seen the counterexample at Sum of Closed Operators Closable?, but I don’t understand why $A$ and $B$ are closed. Could someone expand on this or provide a simpler counterexample? EDIT: Here’s what I’m thinking: If $u_n\to u$ (i.e. $(u_{n,1},\ldots,u_{n,k},\ldots)\to(u_1\ldots,u_k,\ldots)$) $\|Au\|^2=|\textstyle\sum_{k=1}^\infty ku_k|^2+\displaystyle\sum_{k=2}^\infty k^4|u_k|^2$ $\qquad\quad\leq|\textstyle\sum_{k=1}^\infty ku_k|^2+\displaystyle\sum_{k=2}^\infty k^4(|u_k-u_{n,k}|+|u_{n,k}|)^2$ […]

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq (1-\epsilon) \left\| x-y\right\|$ for all $x,y \in \mathbb{R}^n$. Assume that for all $x \in \mathbb{R}^n$, the mapping $a \mapsto f_a(x)$ is continuous. Now let $x_0 \in \mathbb{R}^n$ be […]

Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ Then every compact operator is of almost finite rank: $$\overline{\mathcal{F}(X,E)}=\mathcal{C}(X,E)\subseteq\mathcal{B}(X,E)$$ How do I prove this actual equivalence? Attempt As the image of the unit ball is precompact one has: $$C\in\mathcal{C}(X,E):\quad\|T_NC-C\|=\|T_N-1\|_{C(B)}\to0\quad(T_NC\in\mathcal{F}(X,E))$$ For the converse one might try to […]

Hahn-Banach theorem (second geometric form) exercise #2

Let $X$ be a Hilbert space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F),$$ and any kernel of the involved functionals is not dense on $X$. Using the orthogonality (not Hahn-Banach) in order to prove that there exists scalars $\alpha_1,\alpha_2,\ldots,\alpha_N$ such that $$F\ =\ \sum_{i=1}^N\alpha_iF_i.$$ This is the unanswered last part of […]

Continuity of positive operators

How to prove that an positive linear operator $T:C[0,1]\to R $ in the sense that $T(f)\geq 0$ when $f\geq 0$ is bounded?

If $X$ is separable, then the closed unit ball of $X^*$ is weak-star metrizable. Some calculus helps needed!

Here is my effort to show this fact and I will use ball $X^*$ to denote the closed unit ball of $X^*$. To show ball $X^*$ is weak star metrizable, we only have to show there is a metric $d$ on ball $X^*$ such that the topology induced by $d$ is the weak-star topology on […]

Parseval's Identity holds for all $x\in H$ implies $H$ is a Schauder basis

Prove that any set $\{v_j\}_{j \in \mathbb{Z}}$ for which the Parseval identity $\|x\|^2=\sum_{j=1}^\infty |\langle v_j,x\rangle|^2$ holds for every $x \in H$ is a Schauder basis. I know that a Schauder basis for $X$ if to each vector $x$ in the space there corresponds a unique sequence of scalars $\{c_1,c_2,\dots\}$ such that $x=\sum_{n=1}^\infty c_nx_n$. The conclusion […]

Show that the Newton-Raphson iteration is a contraction under certain conditions

The Newton–Raphson method for finding roots of a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is to iterate the function $$x \mapsto g(x):=x – \frac{f(x)}{f'(x)}$$ Where the following conditions is assumed on $f$ $f$ has a continous second derivative $f'(x) \neq 0 \ \forall \ x \in \mathbb{R}$ There exists some $\alpha \in (0,1)$ such […]