# Tagged Questions

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### Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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### How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
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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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### Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
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### Let $V$be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
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I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ... 1answer 45 views ### The dual of subspace of a normed space is a quotient of dual:$X' / U^\perp \cong U'$I wanted to show that$X' / U^\perp \cong U'$, for$U$being a closed subspace of the Banach space$X$. Therefore I looked at$l: X' / U^\perp \cong U' , x' + U^\perp=[x'] \mapsto x'|_U$. It is ... 1answer 74 views ### Show that a normed Vector space is complete, need smart help. I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ... 1answer 61 views ### Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified? I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ... 5answers 1k views ### Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard? Let's say you are in$\mathbb{R}^n$and you define the norm as$||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product:$||x|| = \sqrt{\langle x, x \rangle}$, ... 2answers 136 views ### Question about Normed vector space. Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ... 0answers 46 views ### Completeness is not preserved under homeomorphism I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$and$\mathbb{R}$) but I have just thought that ... 1answer 55 views ### Show that$\lim_{p \to \infty}||x||_p=||x||_M$[duplicate] Let $$||x||_p=\left( \displaystyle \sum_{i=1}^n|x_i|^p\right)^{\frac{1}{p}}$$ and $$||x||_M=\max\{|x_1|,|x_2|,...,|x_n|\},$$ norms in$\mathbb{R}^n$. Show that $$\lim_{p \to \infty}||x||_p=||x||_M, \ ... 1answer 24 views ### why a lemma shows well-definedness of linear transformations The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation T is well-defined, then if x=y then ... 1answer 54 views ### Banach space problem I came across the following problem: For an open set U in \mathbb{R^n} we define the set of all k-times continuously differentiable functions f:U\rightarrow \mathbb{R} for which D^\alpha f ... 2answers 61 views ### What does this theorem mean? Let (V,\|\cdot\|) be a finite-dimensional normed space. Define \|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}, for all linear operators on V Define \Omega to be the set of all invertible linear ... 0answers 65 views ### The Hilbert space \mathcal{H}_\eta and unitary correspondence with L^2[a,b] The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let \eta(t) be a fixed strictly positive continuous function [a,b]. Define ... 1answer 50 views ### Are C^0[a,b] and C^0[0,1] isometrically isomorphic? Consider C^0[a,b] and C^0[0,1], each equipped with the L^1-Norm. Are these (out of curiosity) isometrically isomorphic? 2answers 108 views ### How does one prove \left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert? Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks! 1answer 79 views ### Are L_p spaces of functions with separable support separable? Let X be a separable space. Is L_p$$(X, \mu, V)$ a separable space? Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$\|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ... 2answers 116 views ### Density of C^{1}_{0}(\mathbb R) in L^{\infty}(\mathbb R) I am looking for a counterexample to C^{1}_{0}(\mathbb R) ( C^1 functions with compact support) is dense in L^{\infty}(\mathbb R)? Is there some easy counterexample showing that this latter is ... 1answer 21 views ### What happens to l_1 if i change coordinate system. Let x =(x_1,\ldots,x_n) \in \mathbb{R}^n and also x=\sum_{i=1}^m t_i u_i, where t_i \in \mathbb{R} and u_i \in \mathbb{R}^n. Is it true that ||x||_1 \geq \sum_{i=1}^m |t_i| ? 1answer 71 views ### Problem in harmonic analysis suppose p be a fixed psitive real number and f is an entire function with$$\lvert f(0) \rvert^p=\int_\mathbb{C}\ \lvert f(z)\exp(-\alpha\lvert z \rvert ^2) \rvert^p dA(z) $$where \alpha ... 2answers 113 views ### Collecting things that are preserved by (isometric) isomorphisms between normed spaces I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ... 1answer 123 views ### Closure of B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\} (NBHM 2005) If A is the closure in \mathcal{C}[0,1] of the set B where$$B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$$Then Which of the following is true? A is closed. A ... 2answers 129 views ### Are the two infima equal? Let R be a ring (not necessarily commutative) or an algebra over some field with a norm defined on it and let I be an ideal in R. Let a,b \in R. Does it hold that \inf_{i,j \in I}\|ab + ai ... 2answers 34 views ### Another question about integrable functions with a transform I am an engineering student, and taking a real analysis course at demand of my advisor, my inexperience in proofs is giving me hard time. I stumbled upon this example, whose proof left as an exercise. ... 1answer 50 views ### Differentiability of function defined as integral form Let H(t)=\int_{\Bbb R}|f(x)+tg(x)|^p\mathrm dx and f,g\in L^p(\Bbb R). Then, how to prove that H is differentiable and find its derivative? I think it's impossible to find it by ... 0answers 17 views ### Limit of L^r norm in lebesgue measure theory [duplicate] Let f\in L^r for some r>0 and \mu (X)=1. Then, prove that \lim_{p\to 0}||f||_p=\exp(\int \log|f|d\mu). This is from chapter L^p spaces, but I don't have any idea. How to make \log? ... 2answers 115 views ### Is there always an injective map from a space in its dual space? Today our teacher said that dual spaces are "big" and told us that this is a consequence by Hahn-Banach's theorem. So I was wondering whether the dual space of a space is always "bigger" or equal ... 1answer 210 views ### X normed linear space separable \Longleftrightarrow \exists K \subset X compact s.t. \overline{ \text{span}\{K\}}= X Let X be a normed linear space. Show that X is separable if and only if there is a compact subset K of X for which \overline{ \text{span}\{K\}}= X I can't figure out how to solve this ... 1answer 38 views ### Lipschitz and derivatives Show that if f is bounded function on E that belongs to L^p_1(E), then it belongs to L^p_2(E) for any P_2 > P_1. I am totally clueless on how to start. Is f an element in a ... 1answer 21 views ### Closure of a set and and an “Open Ball” in a normed space Let X be a normed space and fix t \in X. Set T = \{x\in X : ||x-t|| ≤ r\} and S= \{x\in X : ||x-t|| < r\}. Definition: \mathbf{y}\in\operatorname{Closure}(T) if there exists a sequence ... 1answer 108 views ### Strange mistake in this proof about normed separable vector spaces. I am supposed to prove that an infinite-dimensionale separable normed space X constains a countable subset Y that is linearly independent and dense. There are three hints given: First, prove ... 2answers 73 views ### T be the operator from C[0,1] to C[0,1] defined by Tf = f'+f''. Show that the operator T is unbounded. f \in C[0,1], the space of all continuous, complex-valued functions on [0,1] with supremum norm. \|f\|=\sup_{x\in[0,1]}|f(x)|. Let D be the set of f \in C[0,1] such that the first ... 0answers 64 views ### Help for applying Hahn Banach theorem in this exercise . I want to use this version of Hahn Banach: X real vector space, M a subspace, p a sub-linear mapping and T:M\rightarrow\mathbb{R} linear and T(x)\leq p(x) \forall x \in M Then ... 0answers 110 views ### Operator norm of a convolution Consider the operator on L^2(\Bbb R), f\rightarrow f*g, where g\geq 0 is some L^1 function. Show the operator is a bounded linear operator with operator norm equal to ||g||_1. Showing ... 1answer 90 views ### Calculate the norm of this operator C[0,1]=\{ f : [0,1]\to [0,1], f continuous\} ||f||_\infty=\max_{t\in [0,1]} |f(t)| T:C[0,1]\to C[0,1] defined by$$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$Find ||T|| The usual way to do this ... 0answers 133 views ### Two norms \|\cdot\|_a and \|\cdot\|_b on X, and a function f:X\to Y Fréchet differentiable with one of the norms but not with the other one? There is a theorem that if f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1})  is Fréchet differentiable, then replacing the norms with some equivalent norms \|\cdot\|_{X2} and \|\cdot\|_{Y2} ... 2answers 52 views ### normed space little exercise (X,K) a normed space E is a subspace of X. if \exists x_0 \in X that ||x_0||=d(x_0,E)=1 then show$$||e+\lambda x_0||\geq\frac{||e||}{2}\quad \forall e\in E \quad \forall\lambda\in K 
$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R$ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
### Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$
Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...