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.

learn more… | top users | synonyms

2
votes
0answers
46 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
466 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
153 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
29 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
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
108 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 ...
2
votes
1answer
92 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
301 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
735 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
322 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
340 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
68 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
108 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 ...
3
votes
2answers
414 views

How to show that $C[a,b]$ is infinite dimensional?

How can we give a rigorous proof of the fact that the space $C[a,b]$ of all continuous real (or complex)-valued functions defined on a closed interval $[a,b]$, where $a$, $b$ are any two given real ...
2
votes
1answer
61 views

Is it true that non-equivalent norms share no nontrivial open sets?

Suppose $X$ is a normed linear space with two non-equivalent norms $||\cdot ||_1$ and $||\cdot ||_2$. It is clear that they must generate different topologies, but I was wandering if it were true that ...
0
votes
1answer
1k views

About Banach Spaces And Absolute Convergence Of Seires

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
0
votes
0answers
66 views

How to establish the scalar-multiple and the triangle inequalities as properties of the norm in the completion?

For a normed space $(X, ||.||)$, we can find a Banach space $\hat{X}$ which has a dense suspace $W$ that is isometric with $X$; and this space $\hat{X}$ is unique except for isometries. Now while ...
1
vote
2answers
833 views

How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?

Can we take as a dense subset the collection of all the linear combinations of the vectors of the Schauder basis using the rationals as scalars (or the complex numbers with rational real and imaginary ...
0
votes
1answer
31 views

A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
2
votes
1answer
306 views

A stronger statement of Riesz's lemma

Riesz's lemma state that If $Y$ is a proper, closed subspace of a normed space $X$, then for any $\epsilon>0$, there exists $x$ in the closed unit ball of $X$ such that $d(x,Y)>1-\epsilon$. ...
5
votes
2answers
258 views

Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
5
votes
1answer
231 views

Which of the following sets are open in $C^2[0,1]$? Explain. (Topology on normed spaces)

(a) $A= \{f \in C^2[0,1]:f(x)>0,\parallel f'\parallel_{\infty}<1, |f''(0)|>2\}$ (b) $B= \{f\in C^2[0,1]:f(1)<0,f'(1)=0,f''(1)>0 \}$ (c) $C= \{f\in C^2[0,1]:f(x)f'(x)>0$ for $0\le x ...
12
votes
1answer
512 views

Open Mapping Theorem: counterexample

The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to ...
5
votes
4answers
229 views

Can a norm take infinite value? For example, $\|\cdot \|_1$?

A definition for norm from Wikipedia says Given a vector space $V$ over a subfield $F$ of the complex numbers, a norm on $V$ is a function $p: V → \mathbb{R}$ with the following properties: ...
3
votes
1answer
140 views

$\ell^\infty$ and $\ell^1$

Show that $\ell^\infty$ and $\ell^1$ are normed linear spaces. Solution: Since $\ell^p$ is the collection of real sequences $a=(a_1,a_2, ... )$ for which $\sum_{k=1}^{\infty} |a_k|^p < ...
2
votes
0answers
171 views

Continuous, Bounded Normed Linear Spaces

For $f$ in $C[a, b]$, define $||f||_1 = \int_a^b |f|$. Show that this is a norm on $C[a, b]$. Also, show that there is no number $c\geq0$ for which $||f||_{\max}\leq c||f||_1$ for all $f$ in ...
0
votes
1answer
73 views

Norm of normal Operator A

I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$ My question: What exactly means $\sigma(A)$ and why this is true ? I always thouht the only way to get the ...
2
votes
0answers
143 views

Is it a Banach space? If so what is its dual?

Let $(E_n)$ be a sequence of Banach spaces and $(w_n)$ be a sequence of positive real numbers. For $1\leq p <\infty$ define $\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert ...
2
votes
1answer
113 views

Convergence in norm independent of the choice of the norm.

When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing $\left\Vert X_n - Y_n\right\Vert\rightarrow 0$ as $n\rightarrow 0$ ...
2
votes
2answers
168 views

equivalence of norms and direct sum

Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
5
votes
2answers
239 views

What is the reason norm properties are defined the way they are?

Suppose we have a complex vector space $V$. A norm is a function $f : V \rightarrow \mathbb{R}$ which satisfies (i) $f(x) \ge 0$ for all $x \in V$ (Positivity - Non-Negativity) (ii) $f(x + y) \le ...
8
votes
1answer
198 views

A commutator identity for bounded linear maps and the identity operator of a non-zero normed space is never a commutator

Let $ \mathcal{X} $ be a normed linear space and $ S,T: \mathcal{X} \to \mathcal{X} $ be linear operators such that $ S \circ T- T \circ S=1 $. Show that $ S \circ T^{n+1}- T^{n+1} \circ S=(n+1)T^n ...
1
vote
0answers
51 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
3
votes
0answers
90 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
1
vote
1answer
99 views

Small question regarding norms and Holder conjugates.

I'm trying show that if $p,q$ are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$ Where ...
4
votes
1answer
68 views

Length of a curve in normed spaces

Suppose I have a normed vectorspace $(X,\|.\|)$ and a (differential) path $\gamma:[0,1]\rightarrow X$. Can the Length of the curve be defined as $$L(\gamma)=\int_0^1\|\gamma'(t)\|\text{d}t$$ Or do ...
5
votes
1answer
257 views

Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
5
votes
1answer
90 views

Pseudonormable Product Spaces

I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ...
0
votes
1answer
171 views

Surjective isometry

$(x,Ay):=\sum_{k=1}^{\infty}x_k y_k$ and $A:\ell^1 \to c_0^*$ and $x_k\in c_0, y_k\in\ell^1$ I would like to show A is surjective and isometric. I am not sure about the deifnition of an isometriy, ...
2
votes
1answer
167 views

Quotient norm on $X\backslash M$

I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions: 1) Is the quotient norm a norm on the quotient space X\M ? What I ...
1
vote
1answer
372 views

Simple function approximation of a function in $L^p$

I know that, in general, that any function $f \in L^p(X,\mathcal{M},\mu)$ can be approximated arbitrarily well by a simple function $\sum_{k=1}^n \lambda_k \chi_{E_k}$ where $a_k \in \mathbb{C}, E_k ...
1
vote
1answer
188 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
1
vote
1answer
306 views

Some questions about unbounded linear functionals

Let $E$ be a infinite dimensional normed vector space. 1 - How to define a "not continuous" linear functional $f$ in $E$ such that the set $\ker(f)=\{x\in E:\ f(x)=0\}$ is dense in $E$ but ...
2
votes
1answer
319 views

The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $

How does one show that $(C[0,1], ||.||_{2})$ where $||.||_2=(\int_{0}^{1}|f(t)|^2dt)^{1/2}$ and $C[0,1]$ is the space of all continous function which are mapped from $[0,1]\rightarrow \mathbb{R}$, is ...
2
votes
1answer
70 views

Collection of linear functions

Let $X$ be a Banach space. Let $\{Y_\alpha\}_\alpha$ be normed spaces. Let $\{T_\alpha:X\rightarrow Y_\alpha\}_\alpha$ be an infinite collection of bounded linear functions. Is there a way to create ...
3
votes
1answer
78 views

What is the generator of the disc algebra

The disc algebra, as a set, consists of the functions on the unit disc $D$, which are analytic on the interior of the disc and continuous on its boundary. Its addition and multiplication is obvious. ...