# Tagged Questions

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

33 views

### Banach Tarski proof understanding

Theorem (Banach-Tarski Paradox): The unit ball $\mathbb{D}^3 \subset \mathbb{R}^3$ is equi-decomposable to the union of two unit balls. Proof: Let $\mathbb D^3$ be centered at the origin, and $D^3$ ...
38 views

### Reflexive Banach spaces, compactness

Let $X$ be a reflexive Banach space. Then, consider a linear and compact operator $T \colon X \to X$. Prove that if: $\text{inf} \{ \|Tx\| : x \in X\quad \text{s.t.}\quad \|x\| = 1 \} > 0$, ...
36 views

### Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
41 views

### Corresponding norm from a dual norm?

Let $(X,N_1)$ be a Banach space (separable if necessary) and let $(X^*,N_1^*)$ be its dual space. Here $N_1^*$ denotes the classical dual norm associated to $N_1$. Let $N_2^*$ be an equivalent norm ...
44 views

### $C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
59 views

### Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
152 views

### Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
34 views

### Looking for an (outside $\Bbb R$) application of a certain theorem

I have the following theorem in the lecture notes: Let $E$ be a normed vector space and $\Omega \subset E$ be open and connected, and let $F$ be a Banach space. Let $(f_n)$ be a sequence of ...
26 views

### On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty$, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
25 views

### What is the standard notation for the Banach space of functions that vanish at infinity?

Given a topological space $X$ and a real (or complex) valued function $f$ on $X$, we say that $f$ vanishes at infinity if for any $\varepsilon>0$ there is a compact $K \subseteq X$ such that ...
30 views

### Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
31 views

188 views

### Exists a uniformly convex norm on Banach space satisfying certain condition?

Let $E$ be a Banach space with norm $\|\cdot\|$. Assume that there exists on $E$ an equivalent norm, denoted by $|\cdot|$, that is uniformly convex. Given any $k > 1$, does there exist a uniformly ...
18 views

### Bounding the total variation for Banach lattice

Let $E$ be a Banach lattice and $E_+$ denote its positive cone. For $x_1, \ldots, x_n \in E_+$, is there any way to bound $\sum_1^n\|x_i\|$ with $\|\sum_1^n x_i\|$ without using n? Similaryly, for ...
42 views

64 views

### A normed space is Banach iff its unit sphere is complete [duplicate]

Let $X$ be a non-trivial (other than singleton $x$) normed space. Prove that $X$ is a Banach space if and only if $\{x \in X \mid \|x\| = 1 \}$ is complete.
34 views

### Inverse bounded in a Banach space.

Let $X$ be a Banach space and let $A: X \rightarrow X$ be a bounded linear operator such that $A'(\tilde{X})=\tilde{X}$, show that $A$ has a bounded inverse (on its range). If someone could proof ...
52 views

### Spectrum $\sigma(T)$ of $T:l^1 \to l^1$ given by $T((a_j))=\left( \sum_{j=2}^{\infty} a_j \right) e_1 + \sum_{j=2}^{\infty} a_{j-1} e_j$

I'm considering the bounded linear operator $T$ on $l^1$ (the space of all absolutely convergent complex sequences) given by (with $e_k=(\delta_{kj})_{j=1,2,...}$) T((a_j))=\left( ...
52 views

### Let $E$ and $F$ are Banach spaces,and $A: D(A) \subset E \to F$be closed and densely defined unbounded operator, does $N(A) = R(A^*)$?

Let $E$ and $F$ be two Banach spaces and let $A: D(A) \subset E \to F$ be a closed densely defined unbounded operator. Does it follow that $N(A) = R(A^*)^\perp$? Notation. Let $E$ and $F$ be two ...
50 views

### can a Banach space admit a subsymmetric basis AND a symmetric basis?

Definitions. A (Schauder) basis for a Banach space is called symmetric if it is unconditional and uniformly equivalent to all its permutations. It is called subsymmetric if it is unconditional and ...
38 views

### Hermitian Projections on $C[0,1]$

A projection on a complex Banach space $X$ is said to hermitian if its numerical range is real. Does anyone know an example of an hermitian projection on $C[0,1]_{\mathbb C}=C[0,1]\oplus i C[0,1]$?
51 views

### Range of any projection is closed.

Let $X$ be a Banach space and $P$ a projection. Show that the range of any projection is a closed subspace. Can I use the fact that a Banach space is complete and thus closed and that $P = P^2$ to ...
28 views

### Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...
73 views

### Question surrounding Exercise 3.12 of Brezis, function is convex and l.s.c. for the weak* topology.

Consider Exercise 3.12 of Brezis's Functional Analysis, Sobolev Spaces and Partial Differential Equations. Let $E$ be a Banach space and let $x_0 \in E$. Let $\varphi: E \to (-\infty, +\infty]$ be ...
22 views

### discrete convolution $f*g$ belongs to $\ell_\infty$, i.e. the sup norm is finite

Definitions. Fix any $\phi\in(0,1)$ and $\theta\in(0,1)$, and let us define functions f(n)=\left\{\begin{array}{ll}n^{-\phi},&\text{ if }n\geq 1\\0,&\text{ ...
33 views

### Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...
14 views

### Meaning of amalgamated metric sum of $A_n$’s over $0$ and $d_n$ inherited from $\mathbb{R}^2$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
63 views

### How to proof that a finite-dimensional linear subspace is a closed set

Given a linear space V, a field F, a norm $||.||$ on V and a Base B. How do i proof that the sub-space span{$b_1,b_2,...,b_n$} where $b_i \in B$ is a closed set under the topology that is created ...
107 views

### If $f:X \rightarrow Y$ is a linear isomorphism between $X$ and $f(X)$, then show that there exists a continuous linear map from $Y^*$ onto $X^*$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
81 views

### A proof on the domain of semigroup

Let $T$ be an operator in a Banach space $X$ with the domain $D(T)$ equipped with the graph norm \begin{equation*} \|v\|_T=\|v\|_X+\|Tv\|_X \end{equation*} Assuming $\|v\|_T$ is a norm on $D(T)$, my ...
24 views

### Convex hull of $\{ \Vert x \Vert = 1 \}$ is closed in strictly convex space

I'm trying to show that the convex hull of $\{ \Vert x \Vert = 1\}$ is closed in a strictly convex Banach-space. I don't know how to tackle the problem. Are there any nice characterizations for a ...
87 views

### Corollary of the Hahn-Banach theorem.

Recall the Hahn-Banach Theorem in Normed Vector Spaces: "Let $X$ be a NVS, and let $W$ be a linear subspace of $X$. Then, for any $f_W\in W'$ there exists an extension $f_X\in X'$ such that ...
26 views

### $C^r(K,F)$ as a Banach space for $K$ compact, $F$ Banach space

Let $E$ and $F$ be Banach spaces and $K\subset E$ be compact. I want to understand what the "common definition" (if there is one) of the banach space $C^r(K,F)$ of $r$ times continuously ...
32 views

### If there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach

I have been asked to show that if there exists a bounded operator $T:X \to Y$ with $T^{-1}$ bounded then X is Banach iff Y is Banach. I have shown it for $T$ a linear operator. But I can't use the ...
131 views

### Vector space that can be made into a Banach space but not a Hilbert space

Are there any (real or complex) vector spaces which can be made into a Banach space given a suitable norm, but cannot be given a norm that makes it a Hilbert space? I know that the parallelogram law ...
91 views

### Two Banach spaces, if and only if criterion for range of closed unbounded operator to be closed?

Let $E$ and $F$ be two Banach spaces. Let $A: D(A) \subset E \to F$ be a closed unbounded operator. How do I see that $R(A)$ is closed if and only if there exists a constant $C$ such ...
42 views

### Quotient map $\pi : X \rightarrow X / \mathrm{ker}(A)$ is open for a bounded linear operator $A$

I'd like to show: if $A : X \rightarrow Y$ is a bounded linear operator between Banach-spaces, then $\pi : X \rightarrow X / \mathrm{ker}(A)$ is a open map. I found a proof, which I do not really ...
35 views

### A finite dimensional normed vector space is a Banach Space.

I want to show that a finite dimensional normed vector space X over $\mathbb{K}$ ( $\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$ ) is always complete by using the fact that X is isomorphic to ...
18 views

### sum of uniformly bounded projections acting on a Lorentz sequence space

It is known that for every $k\in\mathbb{N}$ there is $N_k\in\mathbb{N}$ such that every $N_k$-dimensional subspace of $\ell_p$, $1<p<\infty$, contains uniformly complemented copies of $\ell_2^k$ ...
### Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$
$a=a(x), b=b(x)$ are elements of $L^p(\Omega)$, $\Omega$ is bounded open subset of $R^n$. Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?
### Relation between $L^1(\mathbb R)$ and $L^2(\mathbb R)$
I know that if $1\le p<q<\infty$ then $L^p\supset L^q$ and $l^p \supset l^q$. But what is the relation between $L^1(\mathbb R)$ and $L^2(\mathbb R)$? I guess there is no relation, i.e. ...