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

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Positivity of a function in $\mathbb{R}^{n}$

We place ourself in $\mathbb{R}^{n}$. We consider a given increasing function $$ g : \begin{aligned} &\mathbb{R}^{+} \to \mathbb{R} \\ &x \;\;\,\mapsto g(x) \end{aligned}$$ Finally, we ...
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89 views

Computing an induced matrix norm

Assume I have a $n \times n$ matrix and a norm defined as $\|A\| = \max \limits_{x \not = 0}\frac{\|Ax\|}{\|x\|}$, where $\|x\| = \sqrt{\sum x_i^2}$. How can I compute this norm?
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72 views

Vector norms on $\Bbb R^n$

Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ be two vectors in $\Bbb R^n$. Suppose $\left|v_i\right|>|u_i|$ for all $i$. Let $\| \cdot\|$ be any vector norm on $\Bbb R^n$. Is it true ...
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2-norm vs operator norm

I have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the ...
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1k views

Zero “norm” properties

I have seen the claim that the l0-norm ($\|X\|_0$ = support(X)) is a pseudo-norm because it does not satisfy all properties of a norm. I thought it to be triangle inequality, but am not able to show ...
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207 views

Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?

Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$. Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
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146 views

How to show that $\|T\|^2=\|T^*T\|$ for a bounded linear operator $T$?

I need to show that for a bounded linear operator, $T$, on a Hilbert space: \begin{align*} \|T\|^2=\|T^*T\| \end{align*} All I have so far: \begin{align*} \|T^*T\|&=\sup\{|\langle T^*Tf,g \rangle ...
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53 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 $\mathbb R^2$?
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52 views

Question about norms

Denote supnorm by $|x| = \max\{|x_1|,...,|x_n|\}$ where $x \in \mathbb{R}^n$. How can we show that this norm and Euclidean norm satisfies the following inequality? $$ |x| \leq ||x|| \leq \sqrt{n}|...
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1k views

What is the difference between the operator norm and the Euclidean norm?

I am quite confused whether the operator norm is the same as the Euclidean norm (2-norm). I know that :$\left \| A \right \|=\sup_{x\neq 0}\frac{\left \| Ax \right \|}{\left \| x \right \|}$. In ...
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345 views

Exponential norm?

Can a norm "grow exponentially"? Let $||\cdot||_*: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0} $ be a norm such that: $$ \lim_{|x| \rightarrow \infty } \frac{ ||x||_* }{ e^{|x|} } > 0 $$ where $...
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2k views

Inequality involving norm of matrix integral

This question seems basic but I could not find an answer. I have seen the inequality $$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$ where $x(t) \in \mathbb{R}^n$ is a ...
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251 views

Maximum 1-norm of matrix times unit vector

If $A$ is a $p \times p$ matrix, what is $$\max_{||u||_2=1} ||Au||_1 ?$$ I am specifically interested in the case when $A$ is positive definite.
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34 views

Equivalence between trace and Euclidean norm

In the paper "On best approximate solutions of linear matrix equations", there is a very small equivalence I don't know where it comes from. Let $A$ be a matrix (either real or complex), and $\|A\|$ ...
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27 views

Are the norms on a vector space unique?

I was watching an online lecture on bounded linear transformations $$T: \mathcal{C}[a,b] \rightarrow \mathcal{C}[a,b]$$ So the condition for $T$ to be bounded was that for all $f \in \mathcal{C}[a,b]...
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19 views

Is there a way to compare the following two operator norms?

Suppose that $K:C([0,1])\to C([0,1])$ is a continuous operator both with respect to $L^2$ and $L^\infty$ norms. Consider the following operator norm $$\sup_{\|f\|_2\leq 1}\|Kf\|_\infty$$ where $\|.\|...
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53 views

Detailed working of $||f||$

Given $\ell^{\infty} \rightarrow \mathbb R$ defined by $$f(a_1,a_2, a_3, ...)=\frac 1{\sqrt {0!}}a_1 + \frac {-1}{\sqrt {1!}}a_2 + \frac 1{\sqrt {2!}}a_3 +... +\frac {(-1)^{n-1}}{\sqrt {(n-1)!}}a_n$$ ...
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51 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x \...
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30 views

Quadratic form $x^T A x $ is less than $||A|| x^T x$?

Does the following hold? $$x^T A x \leq ||A|| x^T x$$ $A$ is a symmetric positive-definite real-valued matrix, and $x$ is a real-valued vector. Norm is Euclidean.
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48 views

What is the norm of this linear operator?

I need to show that $\|A(g_n)\|_1=b-a-\frac1n$ where $A:L^1([a,b])\to L^1([a,b])$ is given by $A(f)(x) = \displaystyle\int_a^xf(t)~dt$, and $g_n:[a,b]\to\mathbb R$ is given by $$g_n(t)= \begin{cases}n ...
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38 views

Prove $\| \cdot \|_{\infty}$ is well-defined on $\ell ^1$

$1 \leq p < q \leq \infty$ $p=1$ and $q= \infty$ Let $(a_k)_{k \geq 1} \in \ell^1$, then $$\sum_{k \geq 1}|a_k|^1< \infty$$ (*) which means $\sup_{k \geq 1 } |a_k|^1 < \infty$. Hence it is ...
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66 views

Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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48 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
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48 views

Are some of the axioms of a norm of a vector space unnecessary?

I have a homework problem where my task is to find out if some of the axioms of a norm of a vector space are unnecessary, meaning they can be derived from other axioms (I presume from the problem ...
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95 views

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
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33 views

Prove that $\| A \| = \lvert y \rvert$

For all $A \in L(\Bbb R^n, \Bbb R)$ there is a unique $y \in \Bbb R^n$ such that $A\textbf{x} = \textbf{x} \cdot \textbf{y}$. Prove that $\| A \| = \lvert \textbf{y} \rvert$. Hint: Cauchy-Schwarz ...
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$A$ is normal, and $B$ is Hermitian, why does $\left\| \rm{AB} \right\|{\rm{ = }}\left\| \rm{BA} \right\|$?

Let $\left\| . \right\|$ be a unitarily invariant norm on $M_n$. If $A, B ∈ M_n$, $A$ is normal, and $B$ is Hermitian, why does $\left\| \rm{AB} \right\|{\rm{ = }}\left\| \rm{BA} \right\|$?
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$|x+y|=|y+x|$ in a normed group

A normed group $(X,+,|\cdot|)$ is a set $X$ equipped with a group operation $+$ and a function $|\cdot|:X\to\Bbb R$ called a norm such that $|x|=0\iff x=0$ $|x-y|\le|x|+|y|$. From these ...
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76 views

Vector norm inequality [duplicate]

Is it true that for vectors $x$ and $y$ in $\mathbb R^n$ $|\Vert x\Vert -\Vert y\Vert| \ge \Vert x-y\Vert $? Can I simply use the triangle inequality $\Vert x\Vert +\Vert y\Vert \ge \Vert x+y\Vert $ ...
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39 views

If $||A^N|| < 1$ then is ||A||<1?

following a previous question in functional analysis I asked Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $. ...
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47 views

Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
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80 views

Proving $\lVert x \rVert_3$ is smaller than euclidean norm

Is there a simple way to show that $\lVert x \rVert_3\leqslant \lVert x \rVert_2$ for vectors in $\mathbb{R}^2$? I've used the result in another problem but can't figure out where to get it from.
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Proving an induced operator norm equality:

The induced matrix norm is defined by $$||A|| = \sup_{x \ne 0} \frac {||Ax||}{||x||} $$ Show that $$||A|| = \sup_{||x||=1} ||Ax||$$ A is only assumed to be square -- not anything more, e.g., not ...
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87 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the Euclidian ...
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36 views

Is $\left\|A^TA(x-y)\right\| = \left\|A^TA\right\|\times \left\|x-y\right\|$ correct? $A \in \mathbb{R}^{n \times n}$

In the derivation of following, I meet a dumb problem: Note: 1. $\left\|\: \cdot \,\right\|$ is the $l_2$ norm. 2. $A \in \mathbb{R}^{n \times n} $ 3. $x,y \in \mathbb{R}^{n}$ $$\frac{\left\|A^...
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226 views

Upper bound of Frobenius norm of product of matrices.

I'm trying to prove that $||AB||_F\leq||A||_2||B||_F$. As far as I know it isn't a hard problem but I was stuck. Any ideas?
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133 views

Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by…

Suppose $p>0$. Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $\|(x,y)\|=(x^p+y^p)^\frac{1}{p}$ for all $(x,y)\in\mathbb{R}^2$ if and only if $p=2$....
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359 views

Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||...
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126 views

Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
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Norm controls the components of a matrix- proof?

For certain norms (such as the Frobenius norm) it is clear that the norm provides component-wise control - each component is at most as large as the norm in magnitude. How do we establish this for ...
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54 views

Bounding a Symmetric Matrix

Consider the following $n \times n$ matrix $A$, which has 1's on the superdiagonal and subdiagonal and 0's elsewhere, i.e. $$\begin{pmatrix} 0 & 1 & 0 & \cdots & \cdots & \cdots &...
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916 views

Show that the dual norm of spectral norm is Nuclear norm.

Could someone help to understand that the dual norm of spectral norm is Nuclear Norm ? We can focus on the real field. Given a matrix $X \in \mathbb{R}^{mn},$ then the spectral norm is defined by: ...
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How do I find the norm of a matrix?

I have the following matrix below and I would like to find the norm of the matrix. I learned that the norm of a matrix is the square root of the maximum eigenvalue multiplied by the transpose of the ...
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A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
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165 views

Verifying that the Sobolev space is a Banach Space

In PDE Evans 2nd edition (pages 262-263), I am trying to understand a proof of a theorem, which states: THEOREM 2 (Sobolev spaces as function spaces). For each $k=1,\ldots$ and $1\le p\le\infty$, ...
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44 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get $...
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40 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where $l_1,l_2,u_1,u_2\...
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64 views

Is every regular polygon the unit ball for some norm?

For every regular polygon, is there a norm such that the polygon is it's unit ball centered on 0?
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72 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ 2.$\|...
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Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...