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

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Should you drop the inner absolute value sign for $L2$ norm?

Lp norm is defined as: $ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}$ But often time I see people writing: $\left\| \mathbf{x} \right\| _2 := \bigg( ...
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46 views

$A$ 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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41 views

double horizontal bars notation for sets

I am unsure of the meaning of a specific notation. We have a two-dimansional matrix $I_B$ (representing an image after applying Gaussian blurring to it, but that's just background information). Let ...
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25 views

inequality involving norms and integrals

For a square integrable function $f$, is the following true, and if so under what circumstances? \begin{equation} \left\Vert \int_{a}^{b}f\left(t\right)dt\right\Vert _{2}\leq\int_{a}^{b}\left\Vert ...
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1answer
25 views

2-norm of a matrix & 2-norm of a vector

I have a notation issue when I am doing a homework. I am asked to minimize $||Ax - b||_2^2 + \alpha||x||_2^2$. Here, A is a 4 x 5 matrix while b is a 4 x 1 column vector. I suppose Ax - b is also a ...
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45 views

Frobenius Norm inequality

Is there anyway the following inequality can be proved without using SVD and Unitary matrices properties of the norms? $||AB||_F \le ||A||_2 ||B||_F$
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1answer
46 views

Show $\|\mathbb{x}\|_{\infty} \leq \|\mathbb{x}\|_{2} \leq \|\mathbb{x}\|_{1}$

I can't see this on here, so I am going to post my solution and would appreciate if anyone could give me some tips etc. So, $||\mathbb{x}||_{\infty} = max\{|x_j| j\in[1,n]\} = |x_k|$ I have ...
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3answers
54 views

Find $f\in L^2(0,1)$ with $\int_0^1 xf(x)dx = \langle x, f(x)\rangle = 1$ of minimal norm.

I would like to get more hints to the following question. Find $f\in L^2(0,1)$ with $\int_0^1 xf(x)dx = \langle x, f(x)\rangle = 1$ of minimal norm (with the standard norm in $L^2(0,1)$). I figured ...
2
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1answer
41 views

Notion of weak convergence on a normed space without inner product

It seems weak convergence, $x_n \rightharpoonup x$, means that $\displaystyle\lim_{n\to\infty} \langle x_n,x\rangle = \langle x,x\rangle$. Now if we fix the left position of the inner product, then ...
2
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40 views

Find a function to minimize norm.

I have a problem I cannot find a solution to by myself. It goes like this: We have a Hilbert space spanned by the family of functions $\{\sin(x), \cos(x), \sin^2(x), \cos^2(x), \sin(2x)\}$. The scalar ...
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2answers
39 views

Relationship between equivalent norms and ball subsets?

Consider unit balls under norms $\|\cdot\|_i$ and $\|\cdot\|_j$: $$ B(0,1)=\{x\in\mathbb R|\,\,\,\|x\|_i<1 \} $$ $$ \hat B(0,1)=\{x\in\mathbb R|\,\,\,\|x\|_j<1 \} $$ Consider now the ...
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1answer
45 views

Equivalence of definitions of operator norm over general normed vector spaces

A normed module over a general normed ring $(R, |\cdot|)$ is a module with a norm $(V,\|\cdot\|)$ satisfying $\|rx\|=|r|\|x\|$; the norm on the ring is an absolute value in the usual sense, i.e. ...
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0answers
22 views

Prove that $\|\cdot\|$ is a norm

I am in a pickle. I'm given the following info: Let us define a function $f(t)=a\cos(t)+b\sin(t)$ where $a,b\in\mathbb C$. Defin the norm: $$ \|f\|=\|a\cos(t)+b\sin(t)\|=\sqrt{|a|^2+|b|^2} $$ I am ...
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2answers
27 views

Vector space of complex sequences of the form $a_{2n}=n²a_{2n-1}$ for $n=1,2,…$ is closed (making use of the $l_{\infty}$ norm)

I am trying to show that the subspace, say $A$, of $c_0$ (which I use to denote the space of all complex sequences converging to $0$, equipped with the $l_{\infty}$ norm) which contains the sequences ...
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3answers
84 views

Showing that $\|f\|_1$ is not equivalent to $\|f\|_2$

I have the following norms: $$ \|f\|_1=\int_{t_0}^{t_1}\|f(t)\|_2dt $$ $$ \|f\|_2=\sqrt{\int_{t_0}^{t_1}\|f(t)\|_2^2dt} $$ I need to show their non-equivalence, i.e. that there do not exist numbers ...
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1answer
43 views

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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2answers
34 views

Show that norm is induced by a scalar product

Consider $I = [-1,1]$. Let $C(I)$ be the normed space, equipped with norm \begin{align} ||f||_{2} = \left( \int_{-1}^{1} |f(t)|^2 \, dt \right) ^{1/2} \end{align} Show, that norm is induced by a ...
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1answer
59 views

Minkowski functional

Let $(E,\|\cdot\|)$ be a normed vector space over $\mathbb R$ or $\mathbb C$. Then for $K\subseteq E$ the function $x\mapsto \inf\{\alpha > 0: x\in\alpha K\}$ is called a Minkowski functional and ...
2
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1answer
74 views

Demonstration of Cauchy-Schwarz inequality using Minkovski inequality

I intend to demonstrate Cauchy-Schwarz inequality assuming that Minkovski inequality is true $(i)$. Instead of doing the usual proof I want to apply this second inequality somewhere in the resolution, ...
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0answers
51 views

Difference between a measure and a norm in a euclidean space

For example: If we have a line in $\mathbb{R^2}$, would the length of the line be its norm or measure? Could someone please explain the difference? EDIT: Is there any scenario, where "norm" and ...
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1answer
28 views

Examples for permutation invariant norms

I am looking for nice (concrete) examples of permutation invariant norms on $\mathbb{R}^n.$ It is clear that the $\ell_p$ norms do the job. Could you mention me other ones?
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64 views

A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
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2answers
54 views

Time derivative of the distance between 2 points moving over time

let $d_{ij}$ the distance between 2 points in space $p_i$ and $p_j$. These 2 points are moving over time so it is more correct to write them as $p_i(t)$ and $p_j(t)$. $p_i$ and $p_j$ are, at every ...
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1answer
62 views

Derivative of frobenius norm

I am trying to calculate the derivative of an energy function with respect to a vector xx. The energy is given by: $$ψ(A)=∥A−I∥_F^2.$$ Where A is a square matrix with each column as x (a column ...
5
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1answer
167 views

Minimizing the Frobenius norm of a matrix involving the Hadamard product, $\|X(A\odot Y)-S\|_F$

Let $S\in\mathbb{R}^{L\times N}$ and $A\in\mathbb{R}^{M\times N}$ be known and arbitrary. I'd like to solve the following system: \begin{align} \min_{X\in\mathbb{R}^{L\times M},Y\in\mathbb{R}^{M\times ...
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1answer
26 views

Equivalent norms proof

Is the standard $L^2$ norm $$\|u\|^2=(u,u)$$ equivalent to a weighted $L^2$ norm $$\|u\|^2_g =(gu,u)$$ with g>0? If so, how can one prove this?
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Confused about Euclidean Norm

I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality. All the proofs I have referred to ...
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31 views

Maximizing over a p-Frobenius norm ball

Let $p\in [1,\infty]$, $p^*$ be s.t. $1/p+1/p^* =1$ (when $p=1$, $p^*$ is understood to be $\infty$, and vice versa). $||u||_p :=(\sum_{i=1}^n |u_i|^p)^{1/p}$ is the p-norm for vectors in ...
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0answers
18 views

Proof method for non-equivalence of norms?

Suppose I have 3 norms. I need to prove that any two of them are not equivalent. In my situation, proving that (1 and 3) and (2 and 3) are not equivalent is easy, but proving at (1 and 2) are not ...
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Is the integral form of the polarisation identity useful for anything?

It is well-known that the polarisation identity for real vector spaces is $$ \langle a,b \rangle =\frac{1}{2}\sum_{k=0}^1 (-1)^k\lVert a+(-1)^k b \rVert^2, $$ and the complex generalisation is $$ ...
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1answer
28 views

Is it a property of any vector norm on $\mathbb{C}^n$? [closed]

That $||e_1||$ = 1. Where $e_i$ is the standard basis for $\mathbb{C}^n$.
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3answers
62 views

Prove spectral norm compatible with vector norm

Can someone please show me how to prove $||Ax||_2 \leq ||A||_2 ||x||_2$, where $||A||_2$ is the spectral norm and $ A \in \mathbb{R^{n \times n}} $ and $x \in \mathbb{R^n}$. So far I tried to write ...
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2answers
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Continuity of the function $\mathbb{R}^k \to\mathbb{R}: x\mapsto \ln(1+ \lVert x \rVert)$ [closed]

Examine the continuity of the function $f\colon\mathbb{R}^k \to \mathbb{R}$ defined by $f(x) = \ln (1+ \lVert x \rVert)$, where $\lVert\cdot\rVert$ is a norm.
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85 views

Metric on $\mathbb R^2$ in which each sphere B((x,y)r) is an equilateral triangle

Is there a metric on $\mathbb{R}^2$ in which each sphere $B\bigl((x,y),r\bigr)$ is an equilateral triangle centered at $(x, y)$, one of the vertices has the form $(x',y)$ with $x'\geqslant x$? ...
3
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1answer
53 views

Why does for every matrix norm $\lVert \mathbb{I } \rVert \geq 1$ hold?

Why does for every matrix norm $\lVert \mathbb{\cdot }\lVert $ $$\lVert \mathbb{I } \rVert \geq 1$$ hold (where $\mathbb{I }$ is the identity matrix)? I tried to prove it just by the definitions ...
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0answers
87 views

How to decide if a system is ill conditioned when the matrix condition number is very different for different norms?

A linear system Ax=b is said to be ill-conditioned if the condition number (A)of the coefficient matrix A is far from 1. Consider the system $$\begin{align}x_1 = &b_1 \\ x_1+x_2 = &b_2 \\ ...
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1answer
39 views

orthonormal basis question - linear algebra

Verify that $$v_1 = \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right), v_2=\left(\frac 1{\sqrt{3}},\frac{1}{\sqrt{3}}e^{-2i\pi/3},\frac{1}{\sqrt{3}}e^{2i\pi/3}\right), ...
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1answer
60 views

Closed / open set in $\ell^\infty$ metric space.

Let $F$ be the set of all $x$ in $\ell^\infty$ metric space with $x_n =0$ for all but finitely many $n$, then is $F$ closed? or open? or neither? I know that $\ell^\infty$ is the space of all bounded ...
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1answer
43 views

Show whether or not this is a norm.

Show whether $F: \mathbb{R}^3 \rightarrow \mathbb{R}$ defines a normed vector space on $\mathbb{R}^3$. For $(\underline{x}) = (x_1, x_2, x_3)$, $F(\underline{x}) = (\sum_{i=1}^{3} ...
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1answer
81 views

Differentiability of the maximum norm on $\mathbb{R}^{2}$

On $\mathbb{R}^{n}$, we have the norm $\| \cdot \|_{\infty}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{>0}$ which sends $x \mapsto \max{(|x_{i}|)_{1 \leq i \leq n}}$ My calculus is a bit shaky so I ...
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1answer
52 views

Statistical Distance between two points?

I read the concept of statistical distance and understand somewhat. Given the point $P(x_{1},x_{2})$ in 2-D space, the euclidean distance of point P from origin is given by: $d(O,P) = ...
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2answers
40 views

How can I show that this matrix exponential has norm strictly less than one?

Let $t,\sigma,R \in \mathbb{R^+}$. Let $$ \mathrm{A} = \left\lbrack\begin{array}{cc}0 & -\sigma\\ \sigma &-R \end{array} \right\rbrack$$ want to show that the induced $\mathcal{l}_2$ norm of ...
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34 views

uniform continuity with respect to the max norm

Let $\mathbb{R}^2$ be equipped with the norm \begin{align*} \|x\|=\max\{|x_1|,|x_2|\},\quad x=(x_1,x_2)\in\mathbb{R}^2 \end{align*} Let $A:\mathbb{R}^2\to\mathbb{R}$ be given by ...
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1answer
54 views

Is a Field Norm a Norm?

The norm $N(a)$ of an $a$ element of a field extension $K/L$ is the determinant of the matrix representing multiplication by $a$. It has the following properties: $$ N(a b) = N(a)N(b) \\ N(ka)=k^n ...
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0answers
30 views

solving Basis pursuit denoising with nuclear norm regularization

$$ \min_{S,L} \quad \left\| S \right\|_{1} + \left\| L \right\|_{*} $$ $$ \text{s.t.}\quad { \left\| D-MS-L \right\| }_{2}^{2}\le \epsilon $$ S,L,D,M are all matrix, $\epsilon$ is scalar, D and M is ...
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1answer
29 views

Open subset of $\ell^1$

How would I go about showing that the following set $$A =\left\{x \in \ell^1 | x_1 < \sum_{n=1}^\infty x_n2^{-n}\right\}$$ is an open subset of $\ell^1$? (where $\ell^1$ is equipped with the norm ...
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0answers
21 views

Can a p-normed matrix space be embadded to a higher p-normed space

For any $1\leq p\leq\infty$, the vector space $\mathbb{R}^n_p$ with norm $\displaystyle||x||=\left(\sum_{i=1}^n|x_i|^p\right)^\frac1p$. Defined its linear transformation space $M^n_p$ associated norm ...
0
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1answer
36 views

$|u+w| = |u| + |w|$ iff $\langle u,w \rangle =0$.

I was asked what needs to hold such that $|u+w|=|u|+|w|$. Where $u,w \in \mathbb R^n$. Well, first notice that if $|u+w|=|u|+|w|$ then $|u+w|^2 = |u|^2 + 2|u||w|+|w|^2$ if we go by the definition ...
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1answer
48 views

$|x+y|=|y+x|$ 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 ...
0
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1answer
56 views

Guaranteeing Invertibility with Banach Lemma

I'm trying to find an $\epsilon$ for which the Banach Lemma guarantees $I_n + ɛA_n$ is Invertible, where $A_n$ is a matrix of $1$'s, and $I_n$ is the identity matrix, and $n$ can be any dimension. ...