Tagged Questions

2
votes
3answers
78 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
2answers
35 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
37 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
103 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 ...
3
votes
0answers
50 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
59 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
60 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
160 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
51 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
68 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
55 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
93 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
70 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
78 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
92 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
127 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
59 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
41 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
132 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
134 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 ...
4
votes
1answer
139 views

Limit of a sequence in a vector normed space

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \frac{x_1 + ...+x_n}{n} ...
4
votes
1answer
242 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
98 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\| ...
4
votes
1answer
253 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
330 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
106 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
160 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
75 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
63 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 ...
2
votes
2answers
96 views

Complete normed spaces

Let $(X, ||\cdot||)$ be a complete normed space. Let $||\cdot||$ be a norm on $X$, and assume that there are constants $c_{1}$, $c_{2} \in (0,\infty)$ such that: $c_{1}||x-y||\le||x-y||_{0}\le ...
0
votes
2answers
95 views

Problem in normed spaces

Some help with the following would be great. Let $(X,||\cdot||)$ be a normed space. Let $(x_{n})_{n}$ and $(y_{n})_{n}$ be Cauchy sequences in $(X, D)$. Say also that $s_{n} = ||x_{n} + ...
14
votes
3answers
522 views

Intersection between orthogonal complement of a subspace and a set

Consider the normed vector space $E=\mathbb{R}^n$. Define $ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$. Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap ...
1
vote
1answer
145 views

Problem involving a hyperplane and affine subspace II

I am trying to solve this little problem. Suppose you have a normed vector space $E$. Let $H$ be a hyperplane ( $H=\{x\in E: f(x)= \alpha\}$ for some linear functional $f$ and some real number ...
2
votes
1answer
52 views

P-adic “Norm” and scalability criterion

I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ? What I mean ...
2
votes
1answer
396 views

Multiplication operator norm

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
0
votes
2answers
164 views

Why does $\lVert L(x) \rVert \leq \lVert L \rVert\,\lVert x \rVert$?

Why does $\lVert L(x) \rVert \leq \lVert L \rVert\,\lVert x \rVert$? If $L$ is a linear map between Banach spaces $V$ and $W$, why is this true? Also, is this true for $L$ not a linear map? Thanks! ...
1
vote
1answer
69 views

How do you get pointwise convergence in the context of normed spaces [duplicate]

Possible Duplicate: Norm for pointwise convergence Let $V=C([0,1],\mathbf{R})$ be the vector space of continuous real-valued functions on $[0,1]$. Let $(f_n)$ be a sequence in $V$. Then ...
4
votes
1answer
189 views

Characterization of normed vector spaces of finite dimension

I have this problem: Let $E$ be a normed vector space. $S=\{x\in E : ||x||=1\}$. Show that if $S$ is compact then $\dim E$ is finite. This follows directly from the Riesz's lemma, but in the notes ...