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

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Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
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26 views

proof about real sequences

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^{\infty}x_{k}^{2}$ converges. Let $W\subset V$ be the set of rational sequences with a finite number of terms. ...
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57 views

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
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581 views

compute the similarity between two vectors [on hold]

Euclidean distance is a measure that may be used to compute the similarity between two vectors. Given a query $q$ and documents $d_1, \ldots, d_n$, we may rank the documents $\mathcal{D} = ...
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22 views

geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
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16 views

The meaning of notation $\|x - x^*\|$

I was just wondering what $\|x - x^*\|$ in the following equation means: $$B(\epsilon) = \{x : \|x - x^*\|<\epsilon\} $$ Thanks.
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33 views

Dual norm of the matrix L1 norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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716 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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22 views

Proving result on spectral radius

How do I prove that $$\rho(A)=\inf\limits_{\text{operator norms}}\|A\|,$$ $\rho$ being the spectral radius, $A$ being a complex $n\times n$ matrix and operator norms being induced from vector norms by ...
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Norms Induced by Inner Products and the Parallelogram Law

Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
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Product of Norm of Vectors

I have a simple question ... Consider two vectors of same length $\mathbf{x}$ and $\mathbf{y}$. This identity holds: $|\mathbf{x}'\mathbf{y}|$ $\leq$ $\|\mathbf{x}\|$ $\|\mathbf{y}\|$ Then is ...
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45 views

Model $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty $ into standard form

I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ...
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29 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for ||v|| = 0 for nonzero v, ||.|| being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm better than ...
2
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1answer
33 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
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234 views

Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
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44 views

When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
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115 views

How to find x so that $\|A x\| = \|A\| \|x\|$ holds

The subbordinance property of matrix-vector multiplication states that $\|A x\| \le \|A\| \|x\|$ where $\|x\|$ is the norm of vector $x$ and $\|A\|$ is the induced norm of matrix $A$. Many textbooks ...
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1k views

Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?

Stuck, help, please, I am really new to this. I opened the 2-norm and multiplied by $n$, then I am thinking to square both sides. The problem is that I do not know how to open $(x_1 + x_2 + ... + ...
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33 views

Matrix inner product, and operator and trace norm inequality

I have trouble proving the following inequality. Let a matrix $A \in \mathbb{R}^{M \times N}$, and $\sigma_i(A)$ be the i-largest singular value of A. Define the operator norm and the trace norm as ...
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53 views

Proving result on matrix norms

How do I prove that, letting, for $A\in\mathbb{C}^{n\times n}$: $$(a)\quad\|A\|_1=\max\limits_{i=1,\cdots,n}\left(\sum\limits_{j=1}^n|a_{ij}|\right),$$ $$(b)\quad\|A\|_2=\rho^{\frac12}(A^HA),$$ with ...
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Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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31 views

Norms in $\mathbb R^2$ - Strategy to prove that a norm is a norm on a set.

What points should I prove when I am asked to prove that a particular norm, say $||\mathbf x||=||(x,y)||=(|x|^{1/2}+|y|^{1/2})^2$, is a norm in $\mathbb R^2$? P.S I have read about the difference ...
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815 views

Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
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25 views

A problem on norm preserving and angle preserving and their relations.

I want to solve the following problem and finding some difficulties:- I have done the part (a) easily. My problem is in part (b) and (c). In part (b) after calculation I have achieved that ...
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1answer
41 views

How to check if a given piecewise defined function on $\mathbb R^2$ is a norm?

I want to check if the function $\parallel (x,y)\parallel = \left\{ \begin{array}{cc} \sqrt{x^2+y^2} & \mbox{if } xy \geq 0 \\ \max\{\vert x\vert, \vert y\vert\} & \mbox{if } xy < 0 ...
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Physical meaning of norm of a matrix

I know norm of a vector is a length of a vector from origin. So what is the motivation behind defining the norm of the matrix? What is the physical meaning of norm of a matrix? Any help is ...
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evaluating norm of sum of roots of unity

let $l_1,...,l_n$ be roots of unity. I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$. how can I do to prove this?
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upper bound on matrix exponential

I need an upper bound for the following term norm(I-e^(Ax)) in which A is an n*n real matrix ,x is a scalar and I is the unit matrix. is there any upper bound that is zero at x=0? if not what is the ...
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55 views

The space of distributions endowed with the topology of uniform convergence on bounded sets is not Fréchet.

I found a state, that the space of distributions on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Fréchet space. As far as i can ...
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57 views

Show that the sup-norm is not derived from an inner product

I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$ does not come from an inner product (the norm is defined on all bounded and continuous real valued ...
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36 views

Show there exists a Cauchy subsequence

Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in $$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$ Consider in $X$ the scalar product defined by $$(x | y)_0 = ...
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Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
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29 views

Gradient of l2 norm squared

Could someone please provide a proof for the following rule: $$\nabla\|x\|_2^2 = 2x$$ I.E. why is the gradient of the $L_2$ norm square of $x$ equal to $2x$? Thanks
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Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta ...
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Is the norm on a Hilbert space always finite?

If $H$ is a Hilbert space and $x \in H$ then does it follow that $||x|| < \infty$?
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how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?

In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof: $$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$ ...
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30 views

Norm of difference of two squares of matrices

Let $x,y$ be square matrices and $c$ be any scalar. Is it true that $ \Vert x^2 \Vert - c^2 \Vert y^2 \Vert = \Vert x - cy \Vert ^2$? If this is true then I'm done with the proof of a theorem on ...
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41 views

An Orthogonal Projection with Weighted Norm

In the context of solving a convex program via projected gradient descent i am facing the following problem: $$\min_{x\in\mathbb R^2}\lVert x-y\rVert_M^2,\qquad\lVert x\rVert\le1$$ or written ...
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Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
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Seminorm proof of a function

I have an example in a book which is not very clear to me : let $E$ vector space made of numerical functions (or complex) $f$ defined on a set $A$. $\forall a \in A, N_a : f \rightarrow |f(a)|$ is a ...
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$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
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51 views

Are isometries always linear?

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0. Must $f$ be linear in this case ? Note : I am NOT assuming that ...
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31 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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1answer
26 views

Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
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55 views

Norm of a Vector

Suppose $A\inℝ^{n,n}$. We Define $$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$ Show it is a norm. Any thoughts?
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245 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
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Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
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40 views

Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1 $)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1 $) if they induce equivalent metrics. At first I thought this ...
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20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
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1answer
62 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...