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.

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3
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2answers
242 views

$l_1$ equipped with the sup norm is NOT a Banach Space

Prove that $l_1 = \{ x = (x_k)_{k\in\mathbb{N}}\subset \mathbb{R};\ \sum_{k\in\mathbb{N}}\ |x_k| < +\infty \}$ equipped with the norm $\| x\| = \mathrm{sup}_{k\in\mathbb{N}} |x_k|$ is NOT a Banach ...
2
votes
0answers
44 views

Finding an orthornormal basis given a bilinear form

Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
4
votes
0answers
72 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
1
vote
1answer
65 views

$L_{k}^{1}([0,1])$ is a Banach space

Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that ...
1
vote
3answers
48 views

generalization of a normed space

I study analysis and have a problem: I have a normed space for example $(X,M)$ that is not complete, how can I complete the space $X$ with respect to norm $M$? please help me Thanks
4
votes
2answers
195 views

Is the image of every open set under a non-zero discontinuous linear function dense in $\mathbb{R}$?

Given a normed space $V$ over $\mathbb{R}$, is it true that the image of every open set of $V$ under a non-zero discontinuous linear function $V\to\mathbb{R}$ is dense in $\mathbb{R}$? I couldnt prove ...
2
votes
1answer
356 views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
1
vote
0answers
71 views

Define metric on set and products

Let $X$ be set. My question is: if adding point $\ast$ to $X$ to get set $X \cup \{\ast\}$ then on countable product $\prod_{n \in \mathbb N_+} X \cup \{\ast\}$ I found it possible to define metric. ...
1
vote
1answer
98 views

Norms on a vector space over $\mathbb{R}$

In an exercise in the book "Topology and groupoids" the following is asked: Let V be a finite dimensional right vector space over $\mathbb{R}$ ($dim_RV=n)$. Show that any $2$ norms on $V$ are ...
4
votes
0answers
107 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 ...
5
votes
3answers
282 views

If $\{x_n\}$ is a Cauchy sequence in a normed vector space, is $\frac{x_n}{\|x_n\|}$ Cauchy?

Let $\{x_n\}$ a Cauchy sequence in a normed vector space $X$. Is $$y_n = \frac{x_n}{\|x_n\|}$$ another Cauchy sequence in $D = \{x\in X : \|x\| = 1\}$? Remark: The idea is prove that if $D$ is ...
5
votes
1answer
281 views

Distance between a point and closed set in finite dimensional space

Let $X$ be a linear normed space. I need to prove that $X$ is finite dimensional normed space if and only if for every non empty closed set $C$ contained in $X$ and for every $x$ in $X$ the distance ...
4
votes
1answer
257 views

If a normed space $X$ is reflexive, show that $X'$ is reflexive.

If a normed space $X$ is reflexive, show that $X'$ is reflexive. Suppose $X$ is reflexive. Then by definition the Canonical mapping $J : X \to X''$ defined by $x \mapsto g_x$ where $g_x(f) = f(x)$ is ...
6
votes
1answer
148 views

why must a normed space homeomorphic to a complete metric space be complete?

Why must a normed space X homeomorphic to a complete metric space Y be complete? I've solved this in the case where Y is a normed space (considering open balls gives Lipschitz equivalence), but am ...
1
vote
1answer
33 views

closed subspace of $\ell_1$ such that sequences of alternating terms cover $\ell_1$

If $X$ is a closed subspace of $\ell_1$ such that every sequence $y=(x_{2n})\in\ell_1$ can be seen as the 'every other term' sequence given by some $x=(x_n)\in X$, why must there be a constant $C$ ...
1
vote
1answer
176 views

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$. My friends and I have literally been pouring over this problem for days now without success. We've been using Hölder's ...
1
vote
2answers
118 views

Norm of a function, Smoothness Penalization

I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
2
votes
1answer
110 views

finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA
1
vote
1answer
48 views

Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$

Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?
2
votes
1answer
93 views

A specific linear operator between Banach spaces

Let B be the Banach space $B=(C[0,1],\|\cdot\|_{\infty}$) and let $\{\xi_i\}\in l^\infty$. Let $T:l^1\rightarrow B$ be the linear operator given by: $(Ta)(x) = \sum_n\xi_na_nx^n$. I have three ...
3
votes
2answers
199 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
-1
votes
2answers
122 views

Are these sets compact? [closed]

I've some problems concerning this question: Are the following sets compact in $C_{[0, 1]}$ where $ d(x(t), y(t))=\sup_{[0,1]}|x(t)-y(t)|$: $${\{x(t) \mid x(t)=e^{t-a}, a>0\}},~~{ \{ x(t) ...
2
votes
1answer
208 views

Norm of element of Hilbert space as supremum over dense subspace?

Suppose $H_1 \subset H_2$ are both Hilbert spaces with different inner products $(\cdot,\cdot)_{H_1}$ and $(\cdot,\cdot)_{H_2}$. Suppose also that $H_1$ is dense in $H_2$ and that the inclusion is ...
6
votes
2answers
384 views

equivalence of norms

I would like a little help here: I have two defined norms over $C^{1}([0,1])$ : $\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$ $\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$ I already ...
3
votes
1answer
164 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \frac{x_1 + ...+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \frac{x_1 + ...
3
votes
2answers
165 views

convergence of weighted average. proof [duplicate]

It is well known that for any sequence $\{x_n\}$ in a normed space which converges to a limit $x$, the sequence of averages of the first $n$ terms is also convergent to $x$. That is, the sequence ...
1
vote
1answer
108 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
91 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} \ ...
1
vote
2answers
84 views

Inequality regarding weak-* convergence

Let $X$ be a normed linear space, $\psi \in X^{*}$ and $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ a sequence in $X^{*}$. Show that if $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ converges weak-${*}$ to ...
1
vote
1answer
38 views

Maximun norm over the complex sequence

Is $C_0$ (the space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{x_n} =0$ ) is a Banach space relative to the maximum norm ( $\|x\| =max|x_n| $) and pairwise operations ? ...
4
votes
0answers
81 views

Is the result still true if we drop completeness? [duplicate]

I know how to prove the following exercise ( from Folland) : If $X$, $Y$ are Banach spaces. $T:X\rightarrow Y$ is a linear map such that $f\circ T\in\operatorname{dual}(X)$ whenever $f\in ...
1
vote
2answers
64 views

There does not exist $c \ge 0$ such that $\|f\|_{\max} \le c\|f\|_1$

Let $f \in C[a,b]$ and let $\|f\|_1$ be the $\mathcal{L}^{1}$-norm and $\|f\|_{\max} = \max_{x \in [a,b]}|f|$. They are both norms on the given vector space. I want to prove that $\not \exists c \ge ...
0
votes
1answer
435 views

Isometric isomorphism

In the case that $L:B_1 \rightarrow B_2 $ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $||Lx||_{B_1} = ||x||_{B_2} $) can I say that $L\overline{L}= 1 $ is ...
2
votes
0answers
47 views

Do these inequalities imply these inequalities? (Norms and squaring)

Suppose we have the inequalities involving norms $$\lVert f \rVert_{X}^2 \leq C_1(\lVert f \rVert_{Y}^2 + \lVert f \rVert_{Z}^2)$$ and $$\lVert f \rVert_{X}^2 \geq C_2(\lVert f \rVert_{Y}^2 + \lVert f ...
0
votes
3answers
566 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 ...
0
votes
0answers
30 views

Which of the following functions are norms? [duplicate]

For $x=(x_1,x_2)$, which of the following functions on $\mathbb{R}^2$ are norms? a.) $A_1(x) = 7\mid x_1\mid + 3\mid x_2\mid$, b.) $A_2(x) = \text{max}\lbrace\mid x_1\mid^2,\mid x_2\mid^2\rbrace$, ...
0
votes
2answers
221 views

How do I sketch the following norms:

In $\mathbb(R)^2$ sketch $B((1,2),3)$, the open ball of radius $3$ at the point $(1,2)$, with the following norms: the Euclidean norm $\parallel .\parallel_2$, the supremum norm $\parallel ...
0
votes
0answers
32 views

Prove that $\parallel f\parallel_w=\int_a^b\mid f(t)\mid w(t)dt$ is a norm on $C([a,b])$.

Let $w:[a,b]\longrightarrow\mathbb{R}$ with $w(x)\geq c>0$ for some $c \in \mathbb{R}^+$ and all $x \in [a,b]$. Prove that $$\parallel f\parallel_w=\int_a^b\mid f(t)\mid w(t)dt$$ is a norm on ...
0
votes
0answers
47 views

For $(x_1,x_2)$, is $G_1(x)=|x_1|$ a norm in $\mathbb{R}^2$?

For $(x_1,x_2)$, is $G_1(x)=|x_1|$ a norm in $\mathbb{R}^2$? To prove that a function $p$ is a norm we need to prove the following: $p(av) = |a|p(v)$ $p(u + v) \leq p(u) + p(v)$ $p(v)\ge0$, and if ...
2
votes
1answer
132 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
118 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
60 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 ...
2
votes
1answer
97 views

Do we really need Hahn-Banach that much?

In the texts on functional analysis I'm reading right now, the Hahn-Banach theorem is used to prove, among others, those statements (all spaces are over $\mathbb{R}$ or $\mathbb{C}$): Lemma 1: Let ...
1
vote
1answer
395 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 ...
10
votes
1answer
923 views

Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
1
vote
1answer
461 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
178 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
440 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 ...
1
vote
1answer
73 views

Hypervolume of a $N$-dimensional ball in $p$-norm

Suppose I have a N-dimensional ball with radius R in p-norm: $$ \sum_{n=1}^N |x_n|^p = R^p $$ Is there a closed formula for its (hyper)volume? I can't find anything. If there isn't, can we at least ...
2
votes
0answers
118 views

$\ell^0$ and $\ell^{\infty}$ norms

Let $x \in S^{n-1}$ and such that its coordinates $|x_1|\geq \cdots \geq |x_n|$. Under which condition on $\|x\|_0$ the following inequality is true that $$\|x\|_{\infty}\leq \frac{1}{\sqrt ...