Tagged Questions

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Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
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Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
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Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...
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I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
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The modulus of smoothness of $c_0$ by an equivalent norm

Let $(X,\|\cdot\|)$ be a Banach space. For $t>0$, the modulus of smoothness of $\|\cdot\|$ is defined by $\rho_X(t)=\sup\left\{\dfrac{\|x+ty\|+\|x−ty\|}{2}−1:x,y\in S_X\right\}$. We define an ...
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Existence of nontrivial bounded linear operator?

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$
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Comparing two linear functions

Let $X$ be a Banach space and let $h:X\to\Bbb C$ and $f:X\to\Bbb C$ be two bounded linear functions such that if for some $x\in X$ we have $f(x)=0$ then $h(x)=0$. Prove that there exists a ...
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Some questions about subspaces in Banach spaces

I just have a few question about some things in Banach spaces. Let $X$ be a separable, reflexive Banach space with basis $\{e_{i}\}$. Let $X_{n} = \text{span}\{e_{1},...,e_{n}\}$, then consider the ...
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Let $(X,|.|)$ be a Banach space, and $Y$ is a linear subspace of $X$ which is dense in $X$. Now if we have another norm $|.|_Y$ in $Y$ which is comparable to $|.|$ by $$|y|\leq C |y|_Y, \ \ \ \forall ... 2answers 21 views Bounding the distance between L_\infty and L_2 for a continuous function Consider a set of continuous (or even differentiable) functions f_i(x), all defined for x\in [a,b] for i=1\ldots,N. Can one define a uniform constant c (which may depend on f) such that ... 1answer 46 views Equivalence of norms given continuous identity It is known that \parallel \; \parallel_{1} & \parallel \; \parallel_{2} are equivalent norms over X if there are A,B>0 such that A\parallel x \parallel_{1} \leq \parallel x ... 1answer 33 views Does completing a normed space commute with taking quotients? Let X be a normed vector space and Y \subset X a closed subspace. We consider the quotient X / Y and equip it with the quotient norm. Then we may form the completion \overline{X / Y}. We ... 1answer 37 views Frechet derivative of the sup norm function on C[0,1] I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ... 3answers 56 views Counterexamples for Hölder's inequality when p and q are not conjugate. Hölder's inequality shows that, when$$ \frac{1}{p} + \frac{1}{q} = 1,$$and f\in L^p and g\in L^q, then$$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$Is there an example of this ... 0answers 101 views Strongly convex and Frechet differentiable function in reflexive Banach space We first recall two definitions about strong convexity and Frechet differentiability in normed space. Let (X, \|.\|) be a normed space and f:X\rightarrow\mathbb{R} be a function. (a) f is said ... 0answers 46 views Completeness is not preserved under homeomorphism I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ((0,1) and \mathbb{R}) but I have just thought that ... 1answer 54 views Banach space problem I came across the following problem: For an open set U in \mathbb{R^n} we define the set of all k-times continuously differentiable functions f:U\rightarrow \mathbb{R} for which D^\alpha f ... 0answers 39 views Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces. This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let  A  and  B  ... 0answers 35 views Understanding the right assumption about Gateaux derivative in Banach spaces It is a general fact that the notion of Gateaux derivative is not uniform over the mathematical community i.e. someone requires it to be linear and continue other not and require this additional ... 2answers 55 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 25 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 63 views Duality mappings on finite-dimensional spaces I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let V be a separable, reflexive ... 1answer 50 views Are C^0[a,b] and C^0[0,1] isometrically isomorphic? Consider C^0[a,b] and C^0[0,1], each equipped with the L^1-Norm. Are these (out of curiosity) isometrically isomorphic? 2answers 115 views Density of C^{1}_{0}(\mathbb R) in L^{\infty}(\mathbb R) I am looking for a counterexample to C^{1}_{0}(\mathbb R) ( C^1 functions with compact support) is dense in L^{\infty}(\mathbb R)? Is there some easy counterexample showing that this latter is ... 0answers 52 views Difference between unconditional and absolute convergence in Banach spaces One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ... 1answer 75 views The dual space of normed vector space X is isomorphic to the dual of its completion Let \overline{X} be Banach space and X be dense subset of it. Show that dual space of X and dual space of \overline{X} are isomorphic. Why these are isomorphic? I don't know how to prove ... 1answer 32 views Compact embedding Prove that the embedding j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1}) where \|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\} and \|f\|_\infty denotes the supremum norm, ... 2answers 149 views Gateaux and Frechet derivatives and related notions Let X and Y be normed real vector spaces, and f : X \to Y a map. Let's say that: G) f is Gateaux differentiable at x_0 \in X if for all directions v \in X the limit f'(x_0)(v) := ... 1answer 56 views On the completeness of inner product spaces. Let H be a Hilbert space, equipped with an inner product (\cdot,\cdot)_1 and norm \|\cdot\|_1 induced by it. Let (\cdot,\cdot)_2 be other inner product on H and \|\cdot\|_2 the norm ... 0answers 62 views Space of Continuous mappings to metric spaces I want to ask whether some basic result from the space C([0,1],R), where R is the real space carries over to the space C([0,1],E), where (E,\|\cdot\|_E) is a metric space. We know that ... 1answer 139 views If the dual spaces are isometrically isomorphic are the spaces isomorphic? Let X, Y be Banach spaces such that the duals X^\ast and Y^\ast are isometrically isomorphic. Are X and Y necessarily isomorphic? The answer to the question whether X and Y are ... 1answer 40 views How to detect reflexivity of the closure Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue L_p norm is reflexive when 1 < p < \infty, but it is not reflexive for p = 1. How ... 2answers 110 views Collecting things that are preserved by (isometric) isomorphisms between normed spaces I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ... 1answer 149 views Isomorphisms between Normed Spaces Between two normed (linear) spaces there are several notions of isomorphisms: Linear isomorphisms: linear bijective maps Topological isomorphisms: linear homeomorphisms (due to the linearity these ... 1answer 50 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 ... 3answers 162 views Continuous Linear Functional on \ell^{\infty} I'd like help answering two questions. 1) Prove that there is a continuous linear functional on \ell^\infty such that f(e_n)=0 \ \forall n \in \Bbb{N} and f(a)=5 where a=(1,1,1,1,1,1,\ldots). ... 0answers 59 views why is test function space \mathcal{A} complete I am trying to find out, why the space$$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
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 ...