# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

2answers
34 views

### Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
0answers
28 views

1answer
32 views

### Showing a certain map is a norm.

Define $$\|x\|=\sqrt[3]{(|x_1|^2+|x_2|^2)^{3/2}+|x_3|^3}$$ for any $x=(x_1,x_2,x_3)\in \Bbb R^3$. Show that $\|\cdot\|$ is a norm on $\Bbb R^3$. I stuck on showing triangle inequality. I don't know ...
0answers
33 views

### Finding $\|f\|$

Given $f: \ell^2 \rightarrow \mathbb R$ defined by $$f(a_1,a_2,...)=\sum_{n=1}^{\infty} \frac{(-1)^n}{3^{n-1}} a_n$$ Is $f$ a bounded linear functional? If yes, then find $\|f\|$. It is definitely ...
1answer
52 views

### Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
1answer
29 views

### Clarification about completeness of metric spaces

This is probably a very silly question but it bothers me for some time. We define a metric space $X$ to be complete if every Cauchy sequence in $X$ converges to some point in $X$. But any metric ...
0answers
22 views

### Find value of $p$ such that $\sum |b_n|^p$ converges

Let $(X , \|\cdot \|)=(\ell^2, \|\cdot \|_2)$. Let $e_n \in \ell^2$. Fora bounded linear functional $\phi$ on $X$, let $b_n=\phi (e_n)$ for each $n \geq 1$. Find the value of $p \geq 1$ such that the ...
0answers
20 views

### Showing a subspace is closed

Let $X = (\textbf{c}_{0},\|\cdot\|_{\infty})$ and $Y$ be defined as $$Y := \bigg\{ \{x_{i}\} \in \textbf{c}_{0} : \sum_{i=1}^{\infty} \frac{x_{i}}{2^{i}} = 0 \bigg\}.$$ (1) Show that $Y$ ...
2answers
413 views

### Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?
1answer
34 views

### Show that an open linear mapping between normed spaces is surjective

I'd just like to know where to begin. The exact thing to prove: Let $X$ and $Y$ be normed spaces and $R:X\to Y$ is an open linear mapping. Show that $R$ is surjective. And to be clear, neither of the ...
1answer
18 views

### $X,Y$ be real NLS ; $T:X \to Y$ be a linear map such that $\ker T$ is closed ; then does $T$ have closed graph?

Let $X,Y$ be real normed linear spaces and $T:X \to Y$ be a linear map with closed kernel ; then does $T$ have closed graph ? What if we assume arleast one of $X,Y$ to be complete ?
2answers
37 views

### On the dimension of a real Normed Linear Space possessing a certain property

Let $X$ be a real NLS such that for every proper subspace $Y$ of $X$ , $\exists x \in X$ such that $||x||=1$ and $dist (x,Y)=1$ ; then is $X$ finite dimensional ?
1answer
12 views

### $Y$ is a ( closed) proper subspace of a real NLS $X$ such that $dist (x,Y)=1$ for some $x \in X$ with $||x||=1$ ; is $Y$ finite dimensional?

Let $Y$ be a finite dimensional proper subspace of a real NLS $X$ , we know that we can find $x\in X$ ( depending on $Y$) , such that $||x||=1$ and $dist (x,Y):=\{||x-y||:y\in Y\}=1$ . I would like to ...
2answers
21 views

### $X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ?
0answers
20 views

### $f \in \mathcal l^{\infty}{'}$ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty}$ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]

Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty}$ is a sequence with non-negative terms ; then is $f$ continuous ?
1answer
80 views

### $f_1,…,f_n$ be linear functionals on a real vector space $V$, then is there a norm on $V$ which makes every $f_i$ continuous?

Let $V$ be a real vector space, $f_1,...,f_n$ be linear functionals on $V$; then does there exist a norm on $V$ with respect to which each of $f_i$ is continuous? And what if we have infinitely many, ...
1answer
12 views

### Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ?

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ? If $y=x$ then it is easy to see that that's the case . So I thought let $y \ne x$ ; I tried ...
0answers
12 views

0answers
17 views

### Finding norms on a piecewise function

For each $n=1,2,...$ let the function $g_n \in C[0,1]$ be defined by g_n(t)=\begin{cases} 2nt & 0 \leq t \leq 1/2n \\ 2-2nt & 1/2n \leq t \leq 1/n \\ 0 & 1/n \leq t \leq 1\...
2answers
51 views

0answers
39 views

### $Y,Z$ be linear subspaces of a Banach space $X$ ; $Y$ be finite dimensional , $Z$ closed in $X$ ; is $Y+Z$ closed in $X$? [duplicate]

Let $Y$ and $Z$ be linear subspaces of a Banach space $X$ , such that $Y$ is finite-dimensional and $Z$ is closed in $X$ , then is $Y +Z$ also closed in $X$ ?
0answers
26 views

1answer
57 views

### Characterizing orthogonally-invariant norms on the space of matrices

Denote by $M_n$ the space of $n \times n$ real matrices. We say a norm on $M_n$ is orthogonal invariant if: $$\|OX \|=\| XO\|=\|X \| \, \, \forall O \in O_n,X \in M_n$$ I am trying to characterize ...
0answers
26 views

0answers
24 views

### Why is $x\in X$ a weak star continuous linear functional in the dual?

I am reading an excerpt from Infinite Dimensional Analysis by Aliprantis and on page 235 it claims that if $X$ is a normed space, then "$x$ is a weak* continuous linear functional by definition". ...
1answer
44 views

### Is $(1,0,…,0,-n^2,0,0,…) \in \ell^2$?

I am a bit unsure of this because if $n$ is very large then the sum would not be finite but then again the term after the nth term is $0$ onwards.
1answer
46 views

2answers
46 views