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

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How to see if a norm is finite

How do you know if $$||\frac{1}{\sqrt3},\frac{1}{\sqrt8}, ... , \frac{1}{\sqrt{n^2-1}}, ... ||_2$$ is finite. So this means is $$\bigg( \sum _{n=2}^{\infty} | \frac{1}{\sqrt{n^2-1}}|^2 \bigg) ...
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27 views

Bounding a function of norms on the unit cube

For a vector $v \in [0,1]^n$ and $p > 1$ we denote the p-norm of $v$ as: $||v||_p = (\sum_iv_i^p)^{\frac{1}{p}}$. where $v_i$ are the entries of $v$. Define the following (weird looking) function ...
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3 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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9 views

$M_n(A)$ closed in $M_n(A^+)$

Let $A$ be a Banach algebra without unit, and let $A^+$ be the unitization of $A$ equipped with a Banach algebra norm extending the norm on $A$ and such that $||1_{A^+}||=1$. Equip $M_n(A^+)$ with a ...
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15 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
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25 views

Proving polynomial v.s. is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P ∈ X$, define $N_1(P) = \sup_{t∈[0,1]} |P(t)|$ and $N(P) = N_1(P) + |P'(1)|$. I have to prove $N_1$ is a norm ...
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1answer
17 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of ...
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1answer
23 views

L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a ...
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1answer
58 views

Norm restricted to $\mathbb Q$

Consider an algebraic extension $K$ of $\mathbb Q$ and suppose that on $K$ we have a non trivial norm $||\cdot||$. Is it possible to have that the restriction $||\cdot||_{\mathbb Q}$ is the trivial ...
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55 views

Why to introduce norms of vectors?

I am studing Euclidean, metric and normed spaces. What I don't get it is why should I norm a vector. It is usually squared? Why should it be always positive? I've asked this to many people and nobody ...
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28 views

Product of the norms of two vectors w.r.t a symmetric bilinear form

Let $V=V_{n}(q)$ be a $n$ dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $(,)$ be a symmetric bilinear form on $V$. Fix $v\in V$. I would like to show that there exists a ...
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23 views

How to define and compute the norm of a vector with riemannian metric?

Let us consider for example, the riemannian metric $g=e^xdx^2+dy^2$ (it is symmetric and definite positive), with associated matrix $\begin{pmatrix} e^x & 0\\ 0 & 1 \end{pmatrix}$. Consider ...
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+50

Proving Holder's inequality for Schatten norms

Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by $$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$ for $A,B$ $n\times n$ ...
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1answer
24 views

Proving a norm is lipschitz

Let $M\in\mathbb{R}^{n\times n}$. Define the function $f\colon\mathbb{R}^n\to\mathbb{R}$ by $f(x)=\Vert Mx\Vert$. Show that $f$ is Lipschitz. Let $x,y\in\mathbb{R}^n$, then we want to find a ...
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1answer
56 views

Differentiability of norm

Is a norm from any normed space $\mathbb{A}$ to $\mathbb{R}$ Frechet differentiable at zero? I've tried proving by contradiction that the norm is not Frechet differentiable at zero, but didn't get ...
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26 views

If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all $p$ positive integer?

I have got some questions regarding matrix norms and inequalities. We only consider square, nonsingular matrices in the following. If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all ...
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2answers
39 views

Derivative of $l_1$ norm

I want to compute the following derivative with respect to $n\times1$ vector $\mathbf x$. $$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 $$ My work: $$g = \left\lVert \mathbf x - A ...
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1answer
41 views

Not all norms are equivalent in an infinite-dimensional space

How to prove that not all norms are equivalent in an infinite-dimensional vector space? In particular, I would like to prove that for a space $X$ of continuous real-valued functions defined on ...
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1answer
33 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
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11 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
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1answer
19 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
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41 views

Norm of an operator on space of real polynomials

Let $L:\mathbb{R}[X]\rightarrow\mathbb{R}[X]$ be an operator given by the following formula $L\left(\sum\limits_n a_nX^n\right)=\sum\limits_n a_{2n}X^{2n}$. We assume that on $\mathbb{R}[X]$ we have ...
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20 views

Frobenius norm and submultiplicativity

I read (page 8 here) that if $A$ and $B$ are rectangular matrices so that the product $AB$ is defined, then $$(1)\quad||AB||_F^2\leq ||A||_F^2||B||_F^2$$ Does that mean that the inequality above ...
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2answers
25 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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2answers
36 views

Prove norm inequality

It is given that $$\left\lVert x-y\right\rVert =\left\lVert y-z\right\rVert = \left\lVert z-x\right\rVert \qquad (1) $$ where $x,y,z \in \Bbb R^2$ and $ \left\lVert x\right\rVert=\sqrt ...
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22 views

Inequality in Banach space [duplicate]

So I have to either prove or disprove this inequality: $$ \left\lVert x\right\rVert^2 - \left\lVert y\right\rVert^2 \le \left\lVert x-y\right\rVert \left\lVert x+y\right\rVert$$ I know this to be ...
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1answer
32 views

Constructing a specific Rank-One Matrix

Given u $\in \mathbb{R}^{n}$ and v $\in \mathbb{R}^{m}$ with unit $L^{2}$ norm, i.e. $\|u\|_{2}$ = $\|v\|_{2}$ = 1. Construct a rank-one matrix B $\in \mathbb{R}^{mxn}$ such that $Bu = v$ and ...
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Maximum column sum norm of inverse matrix, $\|A^{-1}\|_1$

$A$ is an $N \times N$ nonsingular matrix with bounded maximum row sum norm and unbounded column sum norm, that is, $\|A\|_\infty = O(1)$, and $\|A\|_1=O(N^\alpha)$, where $0<\alpha\leq1$. ...
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36 views

Euclidean norm of two random variables

I have two normally distributed random variables. $X_1$ and $X_2$ with mean $u_1,u_2$ and variance $s_1^2,s_2^2$. They are independent with each other and have interval $(-\infty,\infty)$. Is it ...
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32 views

What is the connection between $l_p$ norms and “$l_p$ metrics”?

In some textbooks metric spaces you sometimes encounter these "$l_p$ metrics", $d_1, d_2, d_\infty$ (I don't think $l_p$ metric is very standard usage) For example, $d_1(x,y) := \sum\limits_i^m ...
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1answer
24 views

Some insight about this integral limit

Given $u:\mathbb{R}^N \rightarrow \mathbb{R}$ is continuous and has compact support, we define the set $$K_u: = \{x\in \mathbb{R}^N : u(x) = \|u\|_\infty\}.$$ Looking at the following limit ...
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3answers
35 views

How do I prove $|\left \langle x,y \right \rangle|=\left \| x \right \|\cdot \left \| y \right \|\Leftrightarrow y=cx,c\in F$

Proving $\Leftarrow$ is easy enough, it's just a matter of plugging it right in. For $\Rightarrow$, I tried changing the right side to $\left (\left \langle x,x \right \rangle \cdot\left \langle y,y ...
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Why can we calculate the supremum of operator norm over unit circle?

I know that to check whether a linear operator is continuous or not we have to check if the operator norm is bounded. $$T: V\to W$$, $$\vert\vert \ T \vert\vert= \sup_{f \in V}\frac{\vert\vert \ Tf ...
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1answer
23 views

Why is the set of units of integer quaternions isomorphic to the quaternion group of order 8?

Let's say that I've got a ring $V$ of integer quaternions of the form $\mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$. Now assume that there exists an element $a = a_1 + a_2i + a_3j + a_4k$ ...
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1answer
53 views

Why does $ab=ba=1$ imply ${a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 = 1$?

Let's say that I've got a group $V$ of integer quaternions of the form $\mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$. Now assume that there exists an element $a = a_1 + a_2i + a_3j + a_4k$ ...
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1answer
36 views

Is the squared euclidean norm a measure for the distance of two points?

I like to prove that a measure for the distance $d$ of two points $\vec a$ and $\vec b$ in $R^N$ is given by the squared euclidean norm $$d^2= \sum^N_j (a_j - b_j)^2 $$ So far I was able to show ...
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1answer
22 views

Supremum Infimum of Norm

Let $A\in\mathbb{R}^{n\times n}$ be an invertible matrix and $\mathbf{x}\in\mathbb{R}^n$. I am trying to prove that ...
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38 views

Corresponding norm from a dual norm?

Let $(X,N_1)$ be a Banach space (separable if necessary) and let $(X^*,N_1^*)$ be its dual space. Here $N_1^*$ denotes the classical dual norm associated to $N_1$. Let $N_2^*$ be an equivalent norm ...
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10 views

Invariance of Frobenious norm under transformation.

Can we say for every invertible square matrix $\mathbf{P}$, $\Vert\mathbf{X-B}\Vert_F^2=\Vert\mathbf{P^{-1}(X-B)}\Vert_F^2$. And does this hold true for non-square matrix $\mathbf{P}$ under some ...
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Comparing Euclidean distances using a vector and its projection

Say we have $x\in\mathbb{R}^n$ and $D\in\mathbb{R}$. Define a corresponding vector $y=[y_1\cdots y_n]$ to be the projection of $x$ onto the $n$-cube of side length $2D$ centered at the origin, i.e. we ...
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1answer
35 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{max}(B^{-1}A)}{\lambda_{min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
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78 views

Showing that the metric $d$ is a norm

Let $X$ be a vector space, and $d:X\times X \to \mathbb{R}$ is a metric on $X$. Also suppose that $d$ is invariant under translations, i.e. $d(x,y)=d(x+z,y+z)$ for all $x,y,z \in X$. Is $d(x,y)$ for ...
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35 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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44 views

Compute the square norm $||\cdot||_2$ of matrix [closed]

Compute the square norm $||\cdot||_2$ of matrix : \begin{bmatrix} O & A_n^{-1} & \dots & O \\ A_n^{-1} & O & \ddots & \\ \vdots & \ddots & \ddots ...
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53 views

What is the example of $L^p$ space which is not a Hilbert Space except $p=2$.

I know that $L^p$-norm satisfy the parallelogram law for $p=2$. But when $p$ is not equal to $2$ then it does not satify the parallelogram law and $L^p$ space is not Hilbert Space. For this I need a ...
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1answer
83 views

Find $||\cdot||_2$ norm of block matrix

Compute the square norm $||\cdot||_2$ of matrix : \begin{bmatrix} O & I_n & \dots & O \\ I_n & O & \ddots & \\ \vdots & \ddots & \ddots & I_n ...
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2answers
30 views

Proving something is a norm

Let $a<b$ be real numbers and $X=C[a,b]$ be the space of continuous functions $f:[a,b] \rightarrow \mathbb R$ Prove that $||f||_1 = \sup _{t \in [a,b]} |f(t)|$ indeed defines a norm on $X$. The ...
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1answer
36 views

Operator Norm: Why is this formulation so rarely seen?

There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...
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1answer
18 views

Show that every norm is a $1$-lipschitz function

Let $\|\cdot\|_0$, a norm on $\mathbb{R}^n$. Show that the function $\|\cdot\|_0$ is $1$-lipschitz and hence, continuous. Meaning, I need to prove that: $$\big|\|x\|_0-\|y\|_0\big| \le \|x-y\|$$ ...
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2answers
38 views

Definition of $L^p$ norm of a vector-valued function

If $u$ is a vector the definition of the discrete norm will be $$\|u\|_{l^p}=(\sum |u_i|^p)^{1/p},$$ If $u$ is a function, $$\|u\|_{L^p}=\left(\int|u|^p\right)^{1/p}$$ But when $u$ is a vector-valued ...