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

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Norm inequality that i cannot show

I have recently stumbled upon an inequality that i can't show \begin{align} \int_a^b\|f(t)-f(b)\|^2dt\leq(b-a)^2\int_a^b\|f'(t)\|^2dt \end{align} None of the tools I know handle the derivative on the ...
3
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2answers
154 views

When does one vector has bigger norm than the other?

Let $u,v\in\mathbb{R}^n$ be two probability vectors (i.e., with all components $\geq 0$ and summing up to 1). Then I was wondering what are necessary and sufficient conditions for $$\|u\|_p\leq ...
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1answer
30 views

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous, then $\|f(x)\|< \infty , x \in X$ and why? [closed]

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous. Then is $\|f(x)\|< \infty , x \in X$ and why? I am thinking yes on this one (because otherwise, it would not be ...
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1answer
56 views

Weakest condition on $f\colon \Bbb R^2\to \Bbb R$ so that $f(\|x\|_1,\|x\|_2)$ is a norm.

$\newcommand{\norm}[1]{\|#1\|_1}\newcommand{\morm}[1]{\|#1\|_2}\newcommand{\xorm}[1]{\|#1\|_3}$ Let $X$ be a finite dimensional Banach space and $f\colon \Bbb R^2\to \Bbb R$. What is the weakest ...
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32 views

Operator norm of $P[v]-P[w]$

Let $\mathcal H$ be a complex Hilbert space with inner product $\langle\mid\rangle$, (dirac notation) which is semi-linear (conjugate linear) in the first argument. Let $\mathcal P_1=\mathcal P_1 ...
0
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1answer
91 views

L2 norm of an inverse of a sum of matrices

I am trying to take the L2 norm of the following expression: $-(H^{-1} + bI)^{-1}v$, where $H$ is a psd matrix, b is a scalar, and $v$ is a vector. In particular I am having trouble with the first ...
6
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2answers
104 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
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1answer
58 views

L1 norm differentiablility

I am trying to understand the Least Absolute Deviation algorithm, which basically is min l1-norm(z) subject to z=Ax-b I want to understand how is the l1-norm ...
3
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1answer
31 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
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32 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
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1answer
13 views

Supremums norm on the open unit ball

Is it correct to say that for a continues function $f$ on $\mathbb{R}^3$ for the supremums norm: $$||f||_{L^{\infty}(\partial B)} \leq ||f||_{L^{\infty}(B)}.$$ That it how is the supremums norm ...
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2answers
39 views

spectral radius

Does the spectral radius of a matrix defines a norm? I mean does it satisfy the properties of norm, ie. $$||x|| \ge 0$$ $$||x|| = 0 \implies x=0$$ $$||kx|| = |k|\;||x||$$ $$||x+y||\le ||x||+||y||$$
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23 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
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1answer
21 views

Equality in definition of dual space norm

In the definition of the dual space norm, the WP page makes the following statement: and I was wondering why going from the middle equality to the right equality was obvious?
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27 views

Function to turn results from a nearest-neighbour function into an inversely proportional version?

Short version: Given an input vector D of n values, what are the different methods that one can use to return a vector W such that each value in W is in inverse proportion to the magnitudes of the ...
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23 views

Norm $p$-adic vector spaces

Are there analogs of euclidean norms such as $\infty$-norm in $p$-adic spaces? What are some of analogies between euclidean space and $p$-adic spaces?
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2answers
122 views

Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...
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2answers
65 views

Proving a theorem about Fourier coefficients

I need to prove this: Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$. Can someone please let me ...
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1answer
37 views

How to show a norm identity of a weighted sum

I ran across the following identity while reading up on norms. It deals with the square of the $2$-norm of a convex combination. That is, for all $x,y,\in\mathbb{R}^{n}$ and $\rho \in [0,1]$: ...
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39 views

When to use which condition number? (which norm)?

The condition number is used to determine how sensitive b is to changes in A in the equation ...
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1answer
80 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
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1answer
41 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
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124 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
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76 views

How to show the equivalence of an Euclidean norm to Euclidean distance?

I have the following problem: I have a set of $N$ vectors. I have some a priori information where the information for each vector comes from. If the two vectors $x_1$ and $x_2$ have been sampled ...
2
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1answer
65 views

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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2answers
65 views

measure theory problems and step functions

I have several questions that I haven't worked out. Any hints or solutions will be appreciated. Suppose that {$f_n$} is a sequence of real valued continuously differentiable functions on [$0,1$] ...
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37 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
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29 views

How to use the condition number to determine whether a matrix is easily inverted?

I have an enormous covariance matrix, but I don't know if it is feasible to take its inverse. So, I thought about finding its condition number, which would help give insight into how easy it might ...
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3answers
400 views

Is there a name for this type of vector norm?

In the case of the $\mathcal{l}_2$ norm we have, $$||\mathbf{x}||_2^2=\mathbf{x}^T\mathbf{x}.$$ I was wondering if there was a type of norm that had a linear operation embedded in it, like this, ...
4
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1answer
425 views

Is the $L_\infty$ norm of the $\mathcal{l}_2$ norm of this sequence of functions finite?

I am interested in proving or disproving the following claim and am stuck. We define a series of functions with the following properties. For each $i\in \mathbb{N}$ let $f_i\colon \mathbb{R}^+ \to ...
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4answers
65 views

Is this a Norm?

How is the following formula calculated, assuming $a$ and $b$ are $n$-dimensional vectors? $\parallel \overrightarrow{a} - \overrightarrow{b}\parallel^2$
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22 views

comparison of norms in vector ODEs

Assume I have the following equation $$ \dot{f}(t) \le A(t)f(t),\quad f(0) = f_0$$ where $f:[0,\infty]\rightarrow\mathbb{R}^n$ has positive components ($\ge0$), ...
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14 views

Prove the following using singular value decomposition

Let $\mathbf{C}\in\mathbb{R}^{m\times n+1}$ have the following singular value decomposition \begin{equation} \mathbf{C} = \mathbf{U}\boldsymbol\Sigma\mathbf{V}^T\\ \mathbf{V} = ...
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2answers
119 views

Does spectral norm of a square matrix equal to its largest eigenvalue in absolute value?

I have one simple question. Given the spectral norm $\left \| . \right \| _2$ of a matrix $A$, which is equal to the squareroot of the largest eigenvalue of $A^{^*}A$ $$\left \| A \right \| ...
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1answer
45 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
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1answer
68 views

How to computer the proximal operator of a indicator function?

For $G_{1}(x) = I_{c}(x), c =\{x|Ax=b\}$, the Proximal operator is: $Prox_{\gamma G_{1}}(x) = Proj_{c}(x) = x + A^{T}(AA^{T})^{-1}(y-Ax)$ I hope to know, how to derivative this result. And, for ...
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1answer
24 views

Comparison of the norms of two non-negative real-valued vectors differing only in one component.

Let $0 \leq a \leq b$ and let $\mathbf{x} \in \mathbb{R^{n}}$. Let $\|.\|$ be a norm over $\mathbb{R}^{n+1}$. If we write $( \mathbf{x},a)$ the vector of $ \mathbb{R}^{n+1}$ made by concatening ...
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1answer
22 views

Can varying $p$ in the $p$-norm induced distanced change which pair of points are closer?

i.e. For some $x, y, z \in \mathbb{R}^{n}$, do there exist $p_{1}, p_{2}$ s.t. $ 0 < \sum_{i=1}^{n} (|x_{i} - y_{i}|^{p_{1}} - |x_{i} - z_{i}|^{p_{1}})$ and $ 0 \ge \sum_{i=1}^{n} (|x_{i} - ...
1
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1answer
34 views

Bound on maximum of product of matrix and vector

I need to bound the absolute maximum of each entry of a matrix-vector product: $\max_{|x|_{1}=1} |Ax|_{\infty}$ I tried to pose this in terms of the induced infinity norm of $A$, as in ...
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42 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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171 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
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1answer
34 views

Operator norm on $M_n(A)$ where $A$ is a Banach algebra

On $M_n(\mathbb{C})$, if we take the operator norm by acting on $\mathbb{C}^n$ where $||(z_1,\ldots,z_n)||=\max_{1\leq i\leq n}||z_i||$, then for $[a_{ij}]\in M_n(\mathbb{C})$ we have ...
3
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1answer
55 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
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39 views

How can we check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
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69 views

Can I solve a problem like a combination of PCA and compressed sensing?

$$ \underset{A,x}{\text{minimize}} \quad \lambda \left\| x \right\|_{1} + \left\| A \right\|_{*} $$ $$ D = A + Mx $$ Where $M \in \mathbb{R}^{n \times m}$, $x \in \mathbb{R}^{m \times z}$, $E=Mx \in ...
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4answers
137 views

Show that $A^{n}\to0$ if and only if $\|A\|^{n}\to0$

Let $\boldsymbol{A}$ be a square matrix. Show that $\lim_{n\to\infty}\boldsymbol{A}^{n}=0$ if and only if $\lim_{n\to\infty}\|\boldsymbol{A}\|^{n}=0$ for the spectral radius or for some operator ...
0
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1answer
46 views

Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ ...
3
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0answers
58 views

Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and ...
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1answer
55 views

Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
1
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1answer
21 views

Matrix factorization inequality

How does one show that the following matrix factorization inequality holds in $M_{n} (\mathcal{A})^{+}$, $$(a_{i}^{*}a^{*}aa_{j}) \leq ||a^{*}a|| \cdot (a_{i}^{*}a_{j})$$ Notation. Let $M_{n} ...