# Tagged Questions

2answers
44 views

### a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
0answers
24 views

### Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
1answer
82 views

### Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
1answer
24 views

### Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
0answers
63 views

### Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
1answer
63 views

### Can natural quotient map between Banach spaces be closed?

Let $X$ be a Banach space and $M$ be closed subspace of X, and let $q:X\to X/M$ be the natural quotient map. I know that $q$ is an open map. I wish to find an example of $X$ and $M$ such that $q$ is ...
0answers
16 views

### About Weakly Lindelöf Determined Banach spaces

I'd like to know where can I read more about weakly lindelĂ¶f determined (WLD) spaces. Especifically, I need to prove: 1.- Every weakly compactly generated is WLD 2.- If X is WLD then (X*,w*) is ...
1answer
44 views

### About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
1answer
56 views

### About open mapping and closed range theorem

I'm self-learning Functional Analysis in Rudin's book and found some following statement hard to understand. Hope someone can help me clarify this. 1) $X, Y$ are Banach spaces, $T \in B(X,Y)$, let ...
1answer
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1answer
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### Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
2answers
156 views

### Do maps between topological spaces somehow induce maps between Banach spaces?

If $X,Y$ are topological spaces and $h:X\rightarrow Y$ is a continuous map, is there some sort of induced map \begin{align*} h':C_b(X)\rightarrow C_b(Y) \end{align*} (or in the other direction) where ...
1answer
169 views

### annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
1answer
310 views

### Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
1answer
50 views

### Family of complemented subspaces

Let $X$, $Y$, $A$, $B$ be topological vector spaces. Given two jointly continuous families of linear injective maps $P: Y \times A \rightarrow X$ and $R: Y \times B \rightarrow X$, such that for $y=0$ ...
0answers
53 views

### Godefroy's Theorem

Anyone knows a proof (books , articles) the Theorem 3.122 (Godefroy) I would like to see other proof using Simons' inequality Any hints would be appreciated.
0answers
163 views

### If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact

Let $X$ be a Banach space and $C\subset X$. $\fbox{1}$ If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact ? $\fbox{2}$ If $C$ is convex , weakly-closed ...
1answer
56 views

### $K$ is weakly-compact $\Longleftrightarrow$ $\Pi(K)$ is weak*-compact

Let $X$ be a Banach space and $K\subset X$. $\displaystyle \Pi:X \longrightarrow X$** canonical injection $\Pi(x)(f)=f(x)$ How can we prove that: $K$ is weakly-compact $\Longleftrightarrow$ ...
1answer
82 views

### The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable

Let $X$ be a Banach space. If $B\subset X$* is a norm-separable How can we prove that: The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable. $X$*$=B(X,\mathbb{R})$ : ...
3answers
164 views

### Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
2answers
240 views

### weak* separable question

(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
2answers
68 views

### Need to confirm: Sup Metric $C[0,1]$, question about boundary

For the sup metric, $C[0,1]$. Let $S \subset C[0,1]$ be given by: $$S=\left\{f:[0,1]\to \mathbb{R} \ : \ 0 \leq f\left(\frac{1}{2}\right)<1\right\}$$ The question is simple: is this set open or ...
1answer
127 views

### Is this set dense in $H^1(\Omega)?$

Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$ dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$ Here $f$ is some linear functional so that $V_1$ is also ...
1answer
98 views

### Is the unit ball of a separable Banach space itself separable?

If $X$ is a separable Banach space, then do we know that its unit ball has a countably dense subset contained in the unit ball? This isn't obvious to me.
1answer
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3answers
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### Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
1answer
155 views

### $c_0[0,1]$ in $C(K)$

Let $K=[0,1]\times \{0,1\}$ be endowed with the topology arising from the lexicographic order on it. It is known that $K$ is compact, Hausdorff, first-countable and perfectly normal. Furthermore, the ...
4answers
338 views

### Does the completeness of a normed vector space only depend on its topology?

Let $V \space$ be a vector space over $\mathbb{R}$, and $\Vert \cdot \Vert_1$, $\Vert \cdot \Vert_2$ norms over $V$, which generate the same topology. Is it always true that if $v_n$ is a Cauchy ...
4answers
872 views

### Example of different topologies with same convergent sequences

It's well known that for metric spaces the following is true Let $X$ be a space with two different metrics $d_1,d_2$ such that the two topological spaces $(X,d_1),(X,d_2)$ have the same ...