Intereting Posts

Prove an identity in a Combinatorics method
Number of ways to express a number as the sum of different integers
An entire function $f(z)$ is real iff $z$ is real
Proving $rank(\wp(x)) = rank(x)^+$
Divisibility question for non UFD rings
Prove that $\frac{d^2y}{dx^2}$ equals $\frac{dy}{dx}×\frac{d}{dy}(\frac{dy}{dx})$
Prove $2^{1/3}$ is irrational.
Matrix identity involving trace
Prove that if $R$ is von Neumann regular and $P$ a prime ideal, then $P$ is maximal
Prove that: $2^n < n!$ Using Induction
Show that $f^{n}(0)=0$ for infinitely many $n\ge 0$.
Relation between Cholesky and SVD
Prove that $5/2 < e < 3$?
Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$
Reconciling Different Definitions of Solvable Group

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.

- Question about Fredholm operator
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- Minimizing continuous, convex and coercive functions in non-reflexive Banach spaces
- Vector space that can be made into a Banach space but not a Hilbert space
- Closure of the span in a Banach space
- Equivalence of reflexive and weakly compact

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- Why unit open ball is open in norm topology, but not open in weak topology?
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- Rainwater theorem, convergence of nets, initial topology
- Nonnegative linear functionals over $l^\infty$
- Volterra Operator is compact but has no eigenvalue
- About Baire's Category Theorem(BCT)
- Distance from a point to a plane in normed spaces
- How common is it for a densely-defined linear functional to be closed?

**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.

- Tensor product definition in Wikipedia
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