Tagged Questions

27 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
30 views

The image of an bounded and linear operator between two banach spaces is closed if the image is finite codimension?

So here is my problem, I would like to prove the following, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded operator. Then $TX$ is closed if it is of finite codimension i.e ...
65 views

37 views

36 views

here we have two cases to study $(1)$ let us fix any $f \in C^{1}[ [0,1] \times [0,1]]$ ($k \neq 0$). Set $$[T(v)](x) := x^{-1}\int_0^xf(x,y)v(y)dy$$ for any $x \neq 0$ otherwise $[T(v)](0) := ... 1answer 68 views Prove that the Set of Bounded Linear Operators is Banach Let$B(V,V')$be the vector space formed by set of linear operators$T:V\rightarrow V'$. where$V,V'$are normed vector spaces. Equip$B(V,V')$with the norm $$\|T\|=\sup\frac{\|T(x)\|}{\|x\|}$$ ... 1answer 35 views Pulling Operator Inside Integral Say$Y$is a Banach space and you have a family of continuous/bounded operators$L_{x}: Y \rightarrow Y$for$x\in \mathbb{R}$and say you have an bounded, smooth map$f(x):\mathbb{R}\rightarrow Y$. ... 2answers 40 views Predual of$\mathcal{B}(K, H)$Is there a predual of$\mathcal{B}(K, H)$? So, what does the space$X$look like, such that$X^*=\mathcal{B}(K, H)$. 1answer 45 views If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication) Let$V,W$be normed vector spaces, and$L(V,W)$be the space of bounded linear operators. Usually I would only see the statement "If$W$is Banach, then$L(V,W)$is Banach.". But Wikipedia writes that ... 0answers 35 views $\gamma-$radonifying operators. I am reading about$\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let$H$be a seperable real Hilbertspace,$E$banach ... 1answer 60 views completely continuous implies compact I'm searching for a proof of the fact that if:$T$is a bounded operator in a reflexive Banach space that maps weakly convergent sequences onto convergent sequences then$T$is compact. If we let ... 0answers 34 views Measurability of the dilatation operator I need some help with this question: We consider the dilatation operator:$T: \mathbb{R^{+}}\to \mathcal{L}(L^p(\mathbb{R}),L^p(\mathbb{R}))\;\;\;\;\;\;\delta\to ...
135 views

$X$ and $Y$ are Banach Spaces.$T$ is a linear bounded operator from $X \to Y$. There exists a real number $c$ which is positive, such that for any $y$ belonging to $T(X)$, there exists a $x$ which ...
59 views

every denting point and strongly exposed point is extreme point

If $X$ be a Banach space and $K$ is a subset of $X$, then I want to prove Every denting point of $K$ is extreme point Every strongly exposed point of $K$ is extreme point $K$ is the closed convex ...
84 views

I'm triyng without success, to find some examples of functions that: $\bullet$Are WOT-measurable, but not SOT-measurable. $\bullet$Are SOT-measurable, but not $||\cdot||$-measurable. I give the ...
229 views

is bounded linear operator necessarily continuous?

Let $U, V$ be separable Banach spaces. Suppose we have a bounded, linear operator $C : U\to V$. Questions are the following *) Shall $C$ be continuous since $V$ is a Banach space? *) In general, ...
66 views

Multiplication operator with a function non-vanishing on the cantor set

Let $M_f$ be the multiplication operator, which acts on bounded functions $g$ on the unit interval as $g\mapsto fg$, with $f:[0,1]\rightarrow \mathbb{C}$ such that $f$ is nonzero only on the Cantor ...
51 views

Why $K(X,Y)$ is closed?

If $X$ and $Y$ are Banach space then $K(X Y)$ is a closed vector space of $B(X, Y)$. $B(X,Y)$ is the vector space of all bounded linear maps from $X$ to $Y$. $K(X,Y)$ is the set of all compact ...
28 views

I'm starting a project about vector- (Banach) valued funtions and measures, and I know some of the basic definitions (measurable,weakly measurable...). The functions of study are ...
81 views

Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
51 views

$B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator

Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$ which of the following is correct? $T$ maps bounded sets into ...
52 views

Reflexivity of $X \times Y$

I want to prove the following Theorem. Let $X,Y$ be reflexive. Then $X \times Y$ is reflexive. Here my try. Proof. Let $J_X, J_Y$ be the canonical injections of $X$ onto $X''$ and of $Y$ onto ...
124 views

Is this proof correct? (left inverse and topologically complementary subsets)

I want to prove the following theorem: Theorem. Assume $T \in \mathcal L ( X, Y )$ is injective. The following statements are equivalent: $T$ admits a left inverse; Im($T$) is closed and ...
131 views

Bounded operators with prescribed range - part II

This is a continuation of the question bellow, in a more particular case. Bounded operators with prescribed range If $X$ is a separable Banach space and $Y$ is a closed, infinite dimensional ...
146 views

Non strictly singular operators

Let $X$ be a separable Banach space and let $T:X\to X$ be a bounded operator that is not strictly singular. Can we always find an infinite dimensional, closed, and complemented subspace $Y$ of $X$ ...
98 views

Bounded operator and dense sets

Let $A : E \rightarrow E$ be a linear operator, and $E$ a (typically infinite-dimensional) Banach space. Suppose that $S$ is a collection of norm 1 elements of $E$ spanning a dense subspace of $E$. ...
236 views

extreme points of the unit ball of the Schatten classes?

Suppose $1<p<\infty$. What are the extreme points of the unit ball of the Schatten classes $S^p$? See below for the definition of $S^p$: http://en.wikipedia.org/wiki/Schatten_norm
144 views

52 views

application of the theorem of the open application

Let $X, Y$ be Banach spaces. Suppose that $T: X \to Y$ is a compact operator. show that if $\dim Y$ is infinite, then $T$ is not surjective. idea: Using the theorem of the open application
215 views

A relatively bounded perturbation of a closed operator is a closed operator.

Please I need help with an example I cant figure out and which will hopefully help me to get the theory: Let $X$ be Banach space and $A, B$ general operators. Furthermore $A$ is closed, ...
221 views

Isometry on a dense sub-space of a Banach space?

Let $X$ be a Banach space and let $D$ be a dense sub-space of $X$. I don't know if the following fact is true: Fact: For every (linear) isometry $T\in\operatorname{Iso}(X)$ and for every ...
93 views

Pitt's theorem and reflexivity

Does it follow from Pitt's theorem that the space of bounded operators from $\ell_2$ to $\ell_p$ ($p<2$) is actually reflexive? We have  \mathcal{B}(\ell_2, \ell_p) = \mathcal{K}(\ell_2, \ell_p) ...
I've got a very straightforward question : if $T : B \rightarrow B$ is a linear continuous operator and $B$ is a Banach space, is $T$ a closed operator? This is obviously true in finite dimension, ...
This is a well-known property of the Hilbert-space adjoint operator that it is WOT continuous. Is a similar true for Banach spaces? That is, for a given Banach space $X$ is the operation ...