6
votes
3answers
45 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 ...
3
votes
0answers
45 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
1
vote
1answer
40 views

What norm on $\mathbb C (z)$

There are several different ways to define a norm on the space of polynomials $\mathbb C [z]$. For example, $\|p\| = \sup_{|z|\le 1}|p(z)|$ defines a norm. If $\mathbb C (z)$ denotes the field of ...
1
vote
1answer
53 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, \ ...
1
vote
1answer
21 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 ...
0
votes
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 $D^\alpha f$ ...
0
votes
2answers
35 views

norm of canonical projection = 1

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set ...
0
votes
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 ...
1
vote
1answer
36 views

Cardinality of maximal subsets with some property

Let $X$ be a normed space. Is it true that all maximal (with respect to "$\subset$") subsets $D\subset X$ with the following property: $$ \|x-y\| \geq1 \textrm{ for } x\neq y, x,y\in D, $$ are of the ...
0
votes
0answers
56 views

Show that the normed space $(l^1, ||.||_1)$ is complete.

I am thinking to start off by saying that $\{x_n\}$ is Cauchy in $l_1$, so for every $\epsilon>0$, there exists an $N$ such that $\sum^\infty_{k=1}\mid x^n_k - x^m_k\mid <\epsilon^2$ for $n$, ...
0
votes
0answers
28 views

Prove line segments are always geodesics in a normed vector space

The length of a path $\gamma: [0,1] \rightarrow \mathbb{V}$ in some normed vector space $\mathbb{V}$ is defined by $$l(\gamma):=sup \sum_{i=1}^n ||\gamma(t_i)-\gamma(t_{i-1})||$$ where the supremum ...
2
votes
0answers
87 views

normed space functional analysis [closed]

applying this lemma for e1=(1,0,0) e2=(0,1,0) e3=(0,0,1) what is maximum of c? i found c=1 am i right?
2
votes
1answer
101 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 ...
0
votes
1answer
99 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
1
vote
1answer
31 views

Let V be a normed vector space. Show that its norm is induced by a scalar product if and only if it satisfies the parallelogramm inequality.

I managed to prove left to right, but I found it hard to get the other direction even if I have the polarization identity. I've found Fréchet - Von Neumann - Jordan theorem that proves exactly what ...
2
votes
2answers
267 views

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 ...
2
votes
1answer
226 views

partial derivatives continuous $\implies$ differentiability in Euclidean space

I am given this theorem: If $f \in C^1(A,\mathbb R^m)$, i.e. every partial derivative of $f$ is continuous on $A$, and $A$ is open in $\mathbb R^n$, then $f$ is differentiable on $A$. Is the ...
0
votes
1answer
737 views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
1
vote
1answer
90 views

Lipschitz condition in infinite dimensional vector spaces

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary ...
0
votes
1answer
91 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
1
vote
1answer
34 views

Distance of a function from a subspace

Let $f \in L^2([-a,a])$. Trying to find $\mathrm{dist}(f,S)$ in $L^2([-a,a])$ (where S is the subspace of real polynomials of max degree $2$, like $a+bx+cx^2$) and knowing that $\langle f,a\rangle=0$ ...
3
votes
4answers
184 views

Show that $c$ is closed in $l^{\infty}$

Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
1
vote
1answer
54 views

Function Spaces

What is exactly the difference between $L^2$ space and ${\ell}^1$ space? I believe that one of them is the space of square of square integrable functions. Does it have to do with one is for series ...
0
votes
1answer
147 views

Convergence in normed spaces

I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$. Does $Q_nf_n$ ...
2
votes
2answers
147 views

If $x\mapsto \| x\|^2$ is uniformly continuous on $E$, the union of all open balls of radius $r$ contained in $E$ is bounded $\forall r > 0$

A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls ...
4
votes
0answers
94 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
1
vote
1answer
95 views

Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$

Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
0
votes
1answer
89 views

Distance of point for a set in linear spaces

Let $X$ a normed linear spaces, $Y \subset X$ a subspace and $z \in X$ an arbitrary point. How can we show that: $$\text{dist} (z, Y) = \sup \{\psi(z) \ | \ \|\psi\| = 1, \psi \equiv 0 \ \text{on} \ ...
0
votes
3answers
472 views

How to find the norm of this bounded linear functional?

Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
2
votes
1answer
119 views

Equivalent statements of continuity of linear operators

I am asked to prove that the following are true: Given a linear operator $T: X \to Y$ where $X,Y$ normed linear spaces: (1) $T$ continuous at at point $\iff$ $T$ continuous everywhere (2) $T$ ...
4
votes
2answers
110 views

Normed linear space and linear functional

Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define $$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$ I think that it ...
1
vote
0answers
58 views

Normed space Analysis

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
1
vote
1answer
307 views

What is the norm of this bounded linear functional?

Let $a$, $b$ be two arbitrary but fixed real numbers such that $a < b$, let $C[a,b]$ denote the normed space of all continuous real (or complex) valued functions defined on $[a,b]$ with the maximum ...
1
vote
1answer
325 views

What are the range and the norm of this bounded linear operator?

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
1
vote
1answer
160 views

How to find the range and inverse of this linear operator?

Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
1
vote
1answer
342 views

About an extension of Riesz' Lemma for normed spaces

The Riesz' Lemma is as follows: Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for ...
4
votes
2answers
222 views

Prove that $(B, \|-\|_{\infty})$ complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions.

Question: Prove that $(B, \|-\|_{\infty})$ is complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions on [0,1]. Context: Old exam problem I'm ...
1
vote
1answer
66 views

find the Fréchet derivative of $F : [0,1] \times \mathcal{C}([0,1]) \rightarrow R : (x,f) \mapsto f(x)$, in $(x_0,f_0)$

As an exercise for my analysis class, I have to find the Fréchet derivative of $F : [0,1] \times \mathcal{C}([0,1]) \rightarrow R : (x,f) \mapsto f(x)$, in $(x_0,f_0)$, where $f_0$ is differentiable ...
1
vote
1answer
46 views

Continuity of $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$

I have a question about a proof in my analysis textbook. They show that if $E$is a banach space, then $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$ is continuous by first showing that it is ...
1
vote
0answers
197 views

equivalence of norms in an open set

I would like to prove that two given norms in the space of smooth functions are equivalent in an open set, is it enough to show that they are equivalent for any compactly contained open set? why? ...
3
votes
1answer
136 views

Prove $\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)}$

How to derive this inequality? $$\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)},$$ where $C$ is constant and ...
5
votes
2answers
311 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
6
votes
1answer
859 views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
2
votes
1answer
113 views

Limit inferior taken on the norm of a sequence

Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| ...
5
votes
1answer
377 views

Norm for continuous linear functionals, newbie questions

Let $E$ be a normed vector space and let $f\colon E \to \mathbb{R}$ be a continuous linear functional. Define the dual norm of $f$ as $$ \|f\| = \sup_{\|x\|\leq 1} |f(x)|. $$ First question. I ...
2
votes
1answer
572 views

Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?

We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, ...
2
votes
0answers
120 views

norms on a vector space - is there a quicker way to approach this problem?

I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to ...
-1
votes
1answer
226 views

strictly convex space ---> strictly convex function

How would you prove that in a strictly convex normed vector space, the function $f(x) = \| x \|^2$ is strictly convex?? FYI: $E$ is strictly convex iff $\| t x + (1-t) y \| <1$ for all $x,y \in ...
2
votes
2answers
105 views

Showing a function is not continuous in the one-norm

I have the following question: Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Let $x_{0}\in [0,1]$ and define $F:C([0,1])\to \mathbb{R}$ by $F(f)=f(x_{0})$ Show that $F$ is ...
2
votes
1answer
76 views

Question about normed spaces

Let $(X,||\cdot||)$ be a complete normed space. Let $F_1, F_2, F_3,\ldots\subseteq X$ be closed, non-empty subsets of $X$. Assume that $F_1 \supseteq F_2\supseteq F_3\supseteq \cdots$ and ...