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

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Why is $L_0$ norm not convex? [closed]

I have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?
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86 views

Find the norm of the operator $A:L_2[0,2] \rightarrow L_2[0,2]$ defined by $(Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)$

I have operator: $\boldsymbol{L}_2[0,2] \to \boldsymbol{L}_2[0,2], ( Ax)( t ) = \boldsymbol{t} \operatorname{sgn}(t-1)x(t)$ I need to find operator norm or say that operator isn't bounded. ...
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106 views

Are the norms $\Vert f\Vert_1=\Vert f\Vert _\infty+\Vert f'\Vert _\infty$ and $\Vert f\Vert _2=\vert f(a)\vert +\Vert f'\Vert _\infty$ equivalent?

We have the norms $\Vert f\Vert_1=\Vert f\Vert _\infty+\Vert f'\Vert _\infty$ and $\Vert f\Vert _2=\vert f(a)\vert +\Vert f'\Vert _\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I ...
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944 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
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69 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
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76 views

When sup norm,i.e. $\| x|| = \sup|x(t)|$ for $\forall t\in T $in C[0,1] for $T \subsetneq [0,1]$

When sup norm, i.e. $\| x|| =\sup|x(t)|$ for $\forall t\in T $in C[0,1] $T$ is such that $T \subsetneq [0,1]$. What condition should be applied to $T$ to make $\| x||$ a norm. I cannot show for ...
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300 views

L1-norm minimization

This is undoubtedly a trivial question but might as well ask: Why is the L1 norm minimization a heuristic for finding the sparsest vector? What I mean is that if the L1 norm sums the elements of a ...
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118 views

Matrix norm equivalence

If we define $ \|A\| = \max \{|A\cdot \mathbf{t}|:|\mathbf{t}|\leq 1\}.$ is it the same as defining it as $\max \{|A\cdot \mathbf{t}|:|\mathbf{t}|= 1\}$ ? If so, why? The book I'm following uses the ...
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79 views

Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
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52 views

Inequality of scalar-product and norm

Why does the following inequality hold, given $A$ is symmetric and $\lambda_{\min} (A)$ is the smallest Eigenvalue of $A$? $$v^\top A v \ge \lambda_{\min} (A) \; ||v||^2$$
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3k views

Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
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989 views

Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v ...
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109 views

Equivalency two norms?

Suppose that the following norms on $C^1[0,1]$ . Are they equivalent norms? $\|f\|=\|f\|_\infty+\|f'\|_\infty$ and $\|f\|=\max\{\|f\|_\infty, \|f'\|_\infty\}$ such that $f\in C^1[0,1]$ , ...
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189 views

What is a decreasing scale of Banach spaces?

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular ...
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1k views

Property of Subordinate Matrix Norm: $\|AB\| \leq \|A\|\|B\|$

I do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown by ...
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181 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
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160 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.
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79 views

Operator norm converging to 0 for certain condition

Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
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974 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 ...
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93 views

Two norms on $C_b([0,\infty])$

$C_b([0,\infty])$ is the space of all bounded, continuous functions. Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$ First I want to prove that it is a norm on $C_b([0,\infty])$. The ...
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87 views

Analysis simple question

Let $S= \{(x_1,\ldots, x_n)\in \mathbb{R}^n$; $|x_1|^p+\ldots+|x_n|^p=1\}$, where $p>1$ is real(and fixed), consider a fixed $y\in\mathbb{R}^n$ and $T:\mathbb{R}^n\rightarrow\mathbb{R}$ such that ...
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37 views

Unique, minimal norm solution of a linear equation

Suppose an equation $Ax=b$ has non-unique solutions. Prove that there exist a unique vector $x_{min}$ satisfying $Ax_{min}=b$ whose norm is the smallest among the solutions of that equation. The ...
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39 views

Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
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43 views

Some inequalities between 1- norm, 2- norm and infinity-norm: $\|x\|_2\leq\sqrt{\|x\|_1\| x\|_\infty}\leq\frac{1+\sqrt{n}}{2}\|x\|_2$

Let $x\in\mathbb{C}^n$. Do the following inequalities hold? $$\lVert x\lVert_2\leq\sqrt{\lVert x\lVert_1\lVert x\lVert_\infty}\leq\frac{1+\sqrt{n}}{2}\lVert x\lVert_2.$$ I think the first inequality ...
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14 views

A complex unital algebra which is a Banach space is also a Banach algebra

In my functional analysis class I got stuck on this question on Banach algebras: Let $ \mathbb{A} $ be a complex unital algebra (it has a unit i.e. a multiplicative identity) and $ \mathbb{A} $ is ...
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14 views

Norm induced by Dot Product?

Prove, that in $[a,b] \to \mathbb{R}$ bounded and closed interval, the continuously differentiable functions(with complex values) on the set $C^1[a,b]$, $$||f||=(\int_a^b |f(x)|^2 dx + \int_a^b ...
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36 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
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41 views

Is $c_{00}$ closed in $(\ell^\infty,\|\cdot\|_∞)$

Consider the normed space $(X,\|\cdot\|)= (\ell^\infty,\|\cdot\|_\infty)$ and its linear subspace $V = c_{00}$ consisting of all sequences $(a_n)_{n≥1}$ of real numbers that eventually become zero: ...
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30 views

Show $F_1$ is a continuous linear functional

Let $(X,||\cdot ||)=(C[0,1],||\cdot ||_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ Show $F_1$ is a continuous linear functional. So we need to ...
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Induced matrix p-norm example

I'm struggling to understand induced matrix p-norms. Our lecture notes have used the example: $$A := \begin{bmatrix}\frac{3}{2} & 0 & \frac{1}{2}\\0 & 3 & 0\\ \frac{1}{2} & 0 & ...
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48 views

Proof related to Hilbert-Schmidt norm

Hi, I am very stuck on this proof and I am not sure how to start it. Any help to get started solving it would be much appreciated.
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30 views

Prove $\sup _{t \in [0,1]} |P(t)|$ is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P \in X$, define $N_1 (P)=\sup _{t \in [0,1]} |P(t)|$ I am having trouble proving the part when you show ...
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31 views

Find the operator norm

Let $T : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation that the matrix $\begin{pmatrix} 1& a \\ 0 &1\end{pmatrix}$ determines. Find the operator norm $||T||$ with respect ...
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16 views

Non convex objective in SVM

In the formulation of svm.. The line underline says the norm of the vector w is a non convex constraint.. But how is this so.. Isn't norm a convex function.. Also aren't the other objectives ...
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31 views

About the definition of $L^{\infty}$ norm

Let $\Omega$ a limited domain in $\mathbb{R}^{n}$, the space $L^{\infty}(\Omega)=\{f: \Omega\to\mathbb{R} $ measurable $; ||f||_{L^{\infty}(\Omega)}<\infty\}$. Then if a function $f \in ...
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37 views

Show that $l^p \subseteq l^q$ for $1 \leq p < q < \infty$

$$l^p = \{ (a_k)_{k \geq 1} : \sum \limits_{k=1}^{\infty} |a_k|^p < \infty \}$$ Since it is said $l^p \subseteq l^q$, I would have thought we have to show $$\sum \limits_{k=1}^{\infty} |a_k|^q ...
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25 views

Why, when $m < n$, does the vector space $S$ of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ always contains a nonzero vector?

Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let S be the set of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ for all $i$. Now I know that $S$ is a vector space. Why is it that when $m ...
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49 views

Proof that a matrix function can be made arbitrarily close to identity

$\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}$ Define the following matrix function: $$ \phi(A) = \sum_{i=0}^\infty \frac{A^i}{(i+1)!} $$ for any $A\in \mathbb{R}^{n \times n}$. A useful ...
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How can I prove that the norm of the following operator is $\frac{1}{m!}$?

I'm trying to prove that the norm of the multilinear symmetric operator $A$ is $\frac{1}{m!}$ where $A$ is defined as: $$ A(x_1,\dots, x_m) = \frac{1}{m!} \sum_{\sigma \in S_m} \xi_1(x_{\sigma(1)} ) ...
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Norm Product expression

Prove the product expression $$\left \| AB \right \|_{U\rightarrow W} \leq \left \| A \right \|_{V\rightarrow W}\left \| B \right \|_{U\rightarrow V}$$ Hint: consider $(AB)u = A(Bu)$ and apply $\left ...
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60 views

the relation between cardinality, L1-norm and L2-norm of a vector

For every $u\in \mathbb{R}^n$, $\textbf{Card}(u)=q$ implies ${\lVert u \rVert}_1 \leq \sqrt{q} {\lVert u \rVert}_2$ where $\textbf{Card}(u)$ is the number of non-zero element (so the L0-norm). Why ...
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36 views

Induced norm of a real matrix, symmetric, positive definite

Describe geometrically $\left\{x\in\mathbb{R}^{n}:\left\Vert x\right\Vert _{A}=1\right\}$ where $\left\Vert \cdot\right\Vert _{A}$ is the induced norm of a real matrix, symmetric, positive definite.
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383 views

Norms and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt. $$ Consider the space $C^1([0,1])$ of ...
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89 views

What is the meaning of $||x||=\sqrt{\langle x,x\rangle}$

I understand that a norm assigns a length to each vector in a vector space. I have been told that $$||x||=\sqrt{\langle x,x\rangle}$$ is a norm. So does this equation find the length of vector $x$, ...
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73 views

Derive the dual function $g(\lambda, \nu)$ for the least-norm problem

I am trying to find the dual function $g(\lambda, \nu)$ to this problem $$\min\limits_{Ax = b} \|x\|$$ Step 1. Form the Lagrangian $$L(x, \lambda, \nu) = \|x\| + \nu^T(Ax-b) = \|x\| + \nu^TAx - ...
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29 views

Why is $\max_{1≤i≤m,1≤j≤n}|a_{i,j}|≤||A||_{op}$

Why is $$\max_{1≤i≤m,1≤j≤n}|a_{i,j}|≤||A||_{op}$$ where $$||A||_{op}=\sup\{||Ax||\space |\space x\in\mathbb{R^n}, ||x||=1\}\space\space\text{(operator norm)}$$ ?
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75 views

soft thresholding derivation with two terms

I am trying to minimize the following function $$argmin_x||x-y||^2_2 + ||x-z||_2^2 + \lambda || \frac{x- w}{c}||_1$$ I have been doing it for while but i am not sure how to do it. I have already been ...
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48 views

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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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 ...