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

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Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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45 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
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31 views

Norm of a real function

Given a function $g:\mathbb{R}^n\to\mathbb{R}$ which is nonnegative, suppose we take any norm of this function. Is it true to say $$\Vert g(x)\Vert =\vert g(x) \vert = g(x)?$$ Additionally, If we ...
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60 views

Norm in finding local extrema for functional

In "The Calculus of Variations" by Bruce van Brunt, he says: Let $J:C^2[x_0,x_1]\to\mathbb{R}$ be a functional of the form $$J(y)=\int_{x_0}^{x_1}f(x,y,y^\prime)dx,$$ where $f$ is a function ...
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66 views

L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value

Consider a matrix $A \in\mathbb R^{n\times m}$ with $n>m$. It has full column rank, i.e. $\operatorname{rank}(A)=m$. Its left pseudo-inverse is given by; $$A^{-1}_\text{left}=(A^TA)^{-1}A^T $$ ...
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59 views

$a_n=\int|f|^nd\mu$ then $\lim_{n\to\infty}a_{n+1}/a_n=||f||_\infty$

Let $(X,\mu)$ be a measure space with $\mu(X)=1$. Let $f$ be a measurable function such that $0<||f||_{\infty}< \infty$. Let $a_n=\int|f|^nd\mu$ Show that: (1) $\lim_{n\to\infty}a_{n+1}/a_n=||...
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The 2-norm of the integral vs the integral of the 2-norm

I`m currently having some issues with a seemingly innocent problem. I would like to show that $$\Bigg|\Bigg|\int_\mathbb{R}\begin{pmatrix}A(x)\\B(x)\end{pmatrix}dx\Bigg|\Bigg|_2 \leq \int_{\mathbb{R}}...
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26 views

Show a bound on the error

Suppose $g \in C[a,b]$, and $M(b-a) < 1$, and let $y$, $y_N$ $\in C[a,b]$ be the unique solutions of the equations, $y = g + Ky$ and $y_N = g + K_N y_N$. $y_N$ has been previously defined as an ...
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52 views

prove absolute integrability given square integrability

am trying to follow the outline of a proof in a book i am reading - must be missing something obvious, but would like to understand what exactly... $f$ is complex and square integrable over e. g. [0, ...
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44 views

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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42 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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1answer
19 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 affine.. ...
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62 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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44 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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52 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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49 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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62 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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53 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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43 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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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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30 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 $L>...
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96 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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27 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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67 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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84 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
51 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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16 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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21 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 \bar{...
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46 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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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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35 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 L^{\infty}...
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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 {x_1^2+x_2^...
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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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37 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 $\|B\|_{...
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61 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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37 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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28 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 $$\lim_{...
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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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27 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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54 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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55 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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44 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 $$\sup_{||\mathbf{x}||=1}||A^{-1}\mathbf{x}||\inf_{||\mathbf{x}||=1}||A\mathbf{x}||=...
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46 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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14 views

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
46 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}) = \frac{\lambda_{\max}(B^{-1}A)}{\lambda_{\min}(B^{...
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85 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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38 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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3answers
56 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 &...