Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
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Operators on $l^{\infty}$

Could you tell me how to find norms of these maps? $$ \phi : l^ {\infty} \ni \{x_n \} \rightarrow \left\{ \frac{x_n}{n} \right\} \in l^2$$ and $$\phi : l^ {\infty} \ni \{x_n \} \rightarrow \left\{ ...
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$\| AB\|_\square \leq 1$ implies $\| BA\|_\triangle \leq 1$

In this post norm denotes a matrix norm, i.e. it is sub-multiplicative. All matrices are real. $A$ is of size $n \times k$ with independent columns ($k \leq n$). $B$ is of size $k \times n$. Let $\| \...
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273 views

What is the matrix norm induced by weighted vector norm?

I denote vector norms with doulbe bars and matrix norms with triple bars. It is well known that the vector norm $L_2$ i.e. $\| x \|_2 = \sqrt{x^\top x}$ induces the matrix norm $||| \cdot |||_2$, ...
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813 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some prime ...
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21 views

Operator in $\mathbb R^2$

I am a bit confused, can someone help me with the following? Is there an operator $T$ in $\mathbb{R}^2$ such that: $\parallel u \parallel +\parallel v\parallel = \parallel T(u+v)\parallel$ for every $...
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70 views

Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho (...
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76 views

Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
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Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
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120 views

Can a vector space be complete for two non-compatible norms?

Let $V$ be a vector space and suppose that we have two non-compatible norms on it, i.e. I distinguish $E = (V, \|\cdot\|_1)$ from $F = (V, \|\cdot\|_2)$ and I ask that $\not\exists C>0 \; \...
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55 views

Calculate $\|f\|$ with $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$

In $C[-1,1]$, consider the norm $\|x\| = \max\{x(t)\mid t \in [-1,1]\}$ and $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$, $x \in C[-1,1]$. Find $\|f\|$ Hi everybody. I got stuck in this problem. I can ...
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1k views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
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357 views

Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: \begin{equation} ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| \end{...
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101 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq n^{p/2}m^...
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89 views

Simple doubt about dual norm

If $(X, \|\cdot\|)$ is a normed vector space, then $$\|F\|_{X^{\prime}}\ =\ \sup_{x\in X-\{0\}}\frac{|F(x)|}{\|x\|},$$ by definition. Then I want prove that, $$\|F\|_{X^{\prime}}\ =\ \sup_{\|x\|=1}|F(...
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563 views

Need help understanding matrix norm notation

I've been trying to understand matrix norms (full disclosure: school assignment, not looking for answers, just clarity!), and how they follow from vector norms - been awhile since I did much linear ...
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554 views

Derive steepest descent vector subject to norm constraint

I am currently working through an old textbook Practical Optimization by Gill, Murray and Wright (c 1982) who make some derivations which seem correct, but I am unable to duplicate. In the equations ...
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175 views

Proving $|x|$ is a norm in $\mathbb {R}^n$

I need some direction on how to start on showing that $| x+y|\leq|x|+|y|$ in $\mathbb R^n$. Note that $$ |x|=\left(\sum\limits_{j=0}^n x_i^2\right)^{1/2} $$ Thank you, Klara
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301 views

Does $N(z)=\pm 1$ imply $z$ is a unit in $\mathbb{Z}[\sqrt{10}]$?

I've been trying to prove that $\mathbb{Z}[\sqrt{10}]$ is not factorial. I did this by defining the norm $N(a+b\sqrt{10})=a^2-10b^2$. I was able to show for myself that $N(z)=\pm 2$ and $N(z)=\pm 5$ ...
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386 views

Finding $L^2$ norm of solution of ODE

I have a linear differential equation with real constant coefficients $$ \sum\limits_{i=0}^3 a_i y^{(i)}(x)=0 $$ with initial conditions $y^{(i)}(0)=y_i\in\mathbb{R}$ where $i=0,1,2$. I need to find $...
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299 views

looking for a norm inequality

I want an inequality of the form : $\Vert a - b \Vert^2 \leq k.(\Vert a\Vert^2 + \Vert b\Vert^2)$ ? where k is a constant. The norm in consideration is the euclidean norm, and $a$ and $b$ are ...
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Proof of Neumann Lemma

Prove that if $\|A\| < 1$, then $I-A$ is invertible. Here, $\|\cdot\|$ is a matrix norm induced by a vector norm. This lemma is referred to as Neumann Lemma. Any ideas on how to go ahead with ...
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672 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
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Subordinate matrix norm

I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
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35 views

If two norms are not equivalent, then we have a sequence $\|x_n\|/\|x_n\|' \to 0$.

Definition 2.22. We say that two norms $\|\cdot\|$ and $\|\cdot\|'$ are equivalent if there exist $C_1,C_2>0$ such that $$C_1\|x_1\|' \le \|x\| \le C_2 \|x\|',$$ for all $x\in V$. If two norms $\|...
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Equivalence of complete norms

Let $\|.\|_1$ and $\|.\|_2$ be two complete norms on a linear space $X$ such that if a sequence $(x_n)$ converges to $x$ in $(X,\|.\|_1)$ and to $y$ in $(X,\|.\|_2)$, then $x=y$. We have to prove that ...
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56 views

Prove that there exists NO norm such that $ f_n $ converges to $f$ iff $f_n$ converges to $f$ on compacta.

More specifically: Given the space of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ where convergence is equivalent to uniform convergence on compacta (i.e., compact sets), prove that ...
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Prove norm inequality

It is given that $$\left\lVert x-y\right\rVert =\left\lVert y-z\right\rVert = \left\lVert z-x\right\rVert \qquad (1) $$ where $x,y,z \in \Bbb R^2$ and $ \left\lVert x\right\rVert=\sqrt {x_1^2+x_2^...
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43 views

Interchange of $\ell^r$ and $L^p$-norm

Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of $L^p$-functions. What is the relation between $\Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p}$ and $\Vert \left(\Vert f_i\Vert_{L^p}\right)_{...
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27 views

Derivative of a nonsingular matrix

Show that : $$\frac{d}{dt} A^{-1}(t) = -A^{-1}(t) (\frac{d}{dt} A(t) ) A^{-1}(t) $$ A(t) is a matrix.
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Why “Re” when squaring norms?

Why does the "Re"-operator pop up in equations when we take a norm and square it? Take, for example, $$\| h - u \|^2 + \|h - w \|^2 = Re\langle h-w, w - u \rangle + \|u - w\|^2 $$ which is taken from ...
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35 views

Linear functions $\mathbb{C}^n\longrightarrow\mathbb{C}^m$ are Lipschitz continous

Exercise: Show that any linear function from $\mathbb{C}^n$ to $\mathbb{C}^m$ is Lipschitz continous. (Hint: Use suitable norms.) I know that the maximum norm, the euclidean norm and the sum norm ...
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The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
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Calculating $A^T A$ in matrix with orthogonal columns

I have a matrix $A$ with three orthogonal columns, and I know that the length (2-norm) of each column is $4$. The question is: what is $A^T A$? Which properties should I use to solve this? Thanks ...
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Prove continuity for a given norm

I struggle with this exercise from an analysis 2 book I use for self study: Let V := $C^1([0,1]; \mathbb{C})$ the vector space of continously differentiable functions from $[0,1]$ to $\mathbb{C}$ ...
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30 views

Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
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Question about an inequality in a published paper which is yielded from an approximation

I am reading a published paper on K- SVD: An algorithm for designing overcomplete dictionaries for sparse representation In the introduction, it says: Recent years have witnessed a growing ...
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Squaring Norms solved by Algebra

I found the following in a paper and am not sure how it is correct: $\Vert a - b \Vert^2$ was expanded to: $\Vert a \Vert^2 - 2a^Tb + \Vert b \Vert^2$ The paper was on gps location algorithms so $...
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53 views

Prove that $(\|x\|^p_X + \|y\|^p_Y)^{1/p}$ is a norm

Let $X$ and $Y$ be normed spaces equipped with the norms $\|\cdot\|_X$ and $\|\cdot\|_Y$, then prove that the following defines a norm on $X\times Y$ for $1\le p < \infty$: $\|(x,y)\| := (\|x\|^...
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80 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
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492 views

a question about symmetric positive definite matrix and norm

If B is $n\times n$ real symmetric positive definite matrix, then $(x,y)=x^TBy$ definites an inner product on $R^n$. How to prove that $||x||=(x^TBx)^{1/2}$ is a norm on $R^n$?
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Norms — Distance Between Vectors

Which two of the vectors $u=(-2,2,1)^T$, $v=(1,4,1)^T$, and $w=(0,0,-1)^T$ are closets to each other in distance for (a) the Euclidean norm? (b) the infinity norm? (c) the 1 norm? I believe I know ...
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Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
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199 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For $...
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Frobenius norm bound

Is there any way to bound Frobenius norm of a product of square matrices A,B and a vector x in the following way: $$ \|ABx\|≤ \|Ax\|\text{ and }\|B\| $$
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67 views

Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i don'...
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655 views

Parallelogram law in normed vectorspace withour an inner product.

Let $V$ be any $\mathbb{K}$-vectorspace with norm $\|\cdot\|$ I know that the Parallelogram law holds if the norm is induced by some inner product $\langle\cdot,\cdot\rangle$, i.e. $$ \newcommand{\...
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167 views

Must vectors in $\mathbb{R}^n$ have their “tail” at origin?

I was looking the definition for an $n$-sphere centered at origin with radius $r$: $$\mathbb{S}^n = \{v \in \mathbb{R}^{n+1} : ||v|| = r \}$$ Although I understand that the $||v|| = r$ condition ...
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67 views

Is there any matrix norm $|| \cdot ||$ such that $||A|| \le ||A||_{\infty} /n$?

Matrix norms are equivalent and can bound each other like some examples on Wikipedia. I was wondering if there is a matrix norm $|| \cdot ||$ that can be upper bounded by $||\cdot ||_{\infty}/n$ ? ...
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Prove that the square sum of eigenvalues is no more than the Frobenius norm for a square complex matrix

Prove: $$ \sum_{r=1}^{n} |\lambda_r|^2 \le \sum_{i,j=1}^{n} |a_{ij}|^2 $$ the equality holds if and only if $\boldsymbol{A^H A=AA^H} $ for a square complex matrix $ \boldsymbol{A}=(a_{ij})_{n\...