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

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Vector norm - Understanding it's geometric meaning in regard to the Euclidean norm

I am trying to understand the vector norm. I have a few subquestions to the primary question here, what is the vector norm? 1. Firstly, lets take the Euclidean norm. Is then $\|x\|=d(x,0)$, where ...
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Examples of infinite dimensional normed vector spaces

In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply ...
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40 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
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16 views

Comparing the Induced Norm of a Matrix Product and its Submatrix Product

Suppose I have a matrix $F \in R^{m \times n}$, and a submatrix of $F$ is defined as $F_S\in R^{|S|\times n}, S\subseteq \{1,2,\dots, m\}$, where $S$ is the subset of row-indices in $F$. What is the ...
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33 views

Dual of Hilbert space : induced norm vs. operator norm

Let $\mathfrak{H}$ be a Hilbert space. Is the operator norm on the dual $\mathfrak{H}^*$ induced by a inner product ?
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41 views

Frobenius norm of product of matrix

The frobenius norm of matrix $F$ with dimension $m\times n$ is defined as $$||F||^2_F = \sum_{i=1}^m\sum_{j=1}^n |f_{i,j}|^2$$ If I have the multiplication of two matrices $$FG$$ where G is matrix ...
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41 views

Condition numbers and block matrices

Assume that $\kappa([A, B])$ is the condition number of a block matrix $[A, B]$. Given that, we also know, $$\kappa(C) < \kappa(A)$$ I am curious whether if the following assertion is true or when ...
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19 views

Minimum of L infinity matrix norm

For a matrix $A \in \mathbb{R}^{m \times n}$, we know that $$ \|A\|_\infty := \sup_{x \neq 0 } \frac{\|Ax\|_\infty}{\|x\|_\infty} = \max_{1\leq i \leq m} \sum_{j=1}^{n} |a_{ij}|.$$ Is there a ...
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32 views

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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17 views

Norm of matrix with denumerable index set

Let $S$ be a denumerable set. Let $\ell_1$ be the set of vectors $x$ indexed by $S$, such that $\sum_{i \in S} |x_i| < \infty$. Let $P$ denote a matrix indexed by $S$ in both dimensions, and let ...
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93 views

Trace minimization subject to diagonal constraints

Problem Revisited - Edited for conciseness: We are given two set of data points X [$p \times n$] and Y [$q \times n$]. Let us assume $X = \hat{X} + \tilde{X}$ and $Y = \hat{Y} + \tilde{Y}$ I am ...
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26 views

(Operator) norm inequality for continuous functions

Let $f,g$ be two non-negative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t,$ $\forall t\in[0,\infty)$. Let be $A$ be a bounded linear operator acting on a Hilbert space. Then I was ...
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18 views

Matrix Inequality : $|| UU^T S + S VV^T -UU^T S VV^T ||_2 \leq 2 ||S||_2$

I am trying to prove the following result: Given any matrix $M \in \mathbb{R}^{m \times p}$, write $\mathcal{P}_M : \mathbb{R}^{m \times p} \to \mathbb{R}^{m \times p}$ for the operator that projects ...
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27 views

When are norms not equivalent?

There are a lot of questions here on showing that two norms are not equivalent. I understand that two norms may not be equivaelent from their proofs, however I do not understand why this happened in ...
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If $\dim X=n$ then for any norm in $X$, $X$ is complete. [duplicate]

I know there are standard proofs for this theorem, but I need to prove it by contradiction or proving that $\dim X=\infty$. I thought maybe using Hahn-Banach? Thanks.
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34 views

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 ...
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146 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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26 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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53 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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29 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 ...
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1answer
48 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 ...
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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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32 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 ...
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1answer
26 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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25 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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10 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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26 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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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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13 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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26 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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11 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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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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60 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
29 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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26 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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62 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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34 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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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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27 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 ...
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1answer
25 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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How can I solve the following exercise [closed]

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
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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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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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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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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, ...
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138 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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60 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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21 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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12 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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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 \| ...