Reflexive Banach Space and Duality
Let $X$ be a Banach space.
1) Show that if $X$ is reflexive, then $X^*$ is reflexive.
Hint: Consider the adjoint operator of the natural embedding $J_X: X \rightarrow X^{**}$.
2) Conversely, show that if $X^*$ is reflexive, then $X$ is reflexive.
Hint: Remark that the natural embedding provides an isomorphism between $X$ and a closed subspace of $X^{**}$.
3) Let $E \subseteq \mathbb{R}^n$ be a measurable set such that $B(0, r) \subseteq E$ for some $r \gt 0$. Use the results from questions 2 and 3 (attached below) to show that $(L^{\infty}(E), \|\cdot\|_{\infty})$ is not reflexive.
Bradz
41
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
Erdos
4.7K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 249 views
- $30.00
Related Questions
- H is a Hilber space
- Prove that if $T \in L(V,W)$ then $ \|T\| = \inf \{M \in \R : \, \|Tv\| \le M\|v\| \textrm{ for all } v \in V \}.$
- Let $f:U\subset\mathbb{R} ^3\rightarrow \mathbb{R} ^2$ given by $f(x,y,z)=(sin(x+z)+log(yz^2) ; e^{x+z} +yz)$ where $U = { (x, y, z) ∈ R^3| y, z > 0 }.$ Questions Inside.
- A constrained variational problem
- A function satifying $|f(x)-f(y)|\leq |x-y|^2$ must be constanct.
- Does the sequence $f_n=\arctan (\frac{2x}{x^2+n^3})$ converge uniformly on $\mathbb{R}$?
- Prove that $p_B :\prod_{\alpha \in A} X_\alpha \to \prod_{\alpha \in B} X_\alpha$ is a continuous map
- Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.
