Intereting Posts

On epsilon delta definition fo Limit.
Bijective local isometry to global isometry
What is the meaning of equilibrium solution?
Is it true that any metric on a finite set is the discrete metric?
Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$
The canonical form of a nonlinear second order PDE
Sequence of numbers with prime factorization $pq^2$
Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?
Cosets of a subgroup do not overlap
Singularity at infinity of a function entire
Fastest Square Root Algorithm
Solving functional equation gives incorrect function
How to prove that $53^{103}+ 103^{53}$ is divisible by 39?
Proof of $\lim_{x \to 0^+} x^x = 1$ without using L'Hopital's rule
A counter example of best approximation

The following question is an exercise and so I’m just looking for advices and not for answers if it’s possible.

I have the following sets in $l^\infty$

$$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} \subseteq c := \{x_n \in l^\infty: \exists \lim x_n\}.$$

And I intend to prove that they are not isometrically isomorphic. I suppose that the problem is that for every $(x_n) \in c_0$ there exists a finite and non-empty set $\{x_{n_i}\}$ such that $|x_{n_i}| = \|x_n\|$ but this is false in $c$, but I don’t know how to continue with this idea. Can you help me?

An interesting fact about this two spaces is that although they aren’t linearly isometric they are linearly homeomorphic given by $T:c \to c_0, T(x_n) = (\lim x_n, x_n-\lim x_n)$. This is too strange to me.

- Weak net convergence in $\ell_p$, where $1 < p < \infty$.
- The reflexivity of the product $L^p(I)\times L^p(I)$
- In a normed space, the sum of a Closed Operator and a Bounded Operator is a Closed Operator.
- Weak limit of an $L^1$ sequence
- $C$ is NOT a Banach Space w.r.t $\|\cdot\|_2$
- Isometry from Banach Space to a Normed linear space maps

- Sets $f_n\in A_f$ where $f_{n+1}=f_n \circ S \circ f^{\circ (-1)}_n$ and operator $\alpha(f_n)=f_{n+1}$
- Matrices A+B=AB implies A commutes with B
- Closed subspace $M=(M^{\perp})^{\perp}$ in PRE hilbert spaces.
- Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C$
- Definition of resolvent set
- First theorem in Topological vector spaces.
- Are there necessary and sufficient conditions for Krein-Milman type conclusions?
- Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$
- Limit of an average integral?
- Monotonicity of $\log \det R(d_i, d_j)$

**Hints:**

1) Prove that unit ball of $c$ have a lot of extreme points. In fact there are $\mathfrak{c}$ extreme points but this is not important for the solution.

2) Prove that unit ball of $c_0$ have no extreme points.

3) Prove that if $x$ is an extereme point of unit ball of some normed space $X$, and $i:X\to Y$ is an isometry, then $i(x)$ is an extreme point of unit ball of $Y$.

4) The rest is clear.

- Cardinality of the set of differentiable functions
- Matrix Calculus in Least-Square method
- Proof that SAT is NPC
- Why are there 12 pentagons and 20 hexagons on a soccer ball?
- Equal elements vs isomorphic elements in a preoder
- Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?
- Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?
- Interesting limit involving gamma function
- Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.
- Intersection between two planes and a line?
- Discrete logarithm – strange polynomials
- Total Time a ball bounces for from a height of 8 feet and rebounds to a height 5/8
- How to arrive at Ramanujan nested radical identity
- If $x\in \left(0,\frac{\pi}{4}\right)$ then $\frac{\cos x}{(\sin^2 x)(\cos x-\sin x)}>8$
- $L^p(\mathbb{R})$ separable.