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

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staircase length in Whitney's flat norm and Jenny Harrison's natural norm

Can someone provide the complete calculation for the length of a staircase as it converges to a diagonal line in Euclidean space in a sequence in which the number of steps goes to infinity between two ...
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29 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=Sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
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48 views

The choice of scalar factors in the proof of the Schwarz inequality

In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ and ...
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47 views

Upper bound on infinity norm of inverse of a positive definite matrix

Consider a positive definite matrix, $A$, and the following quantity: \begin{align} \|A^{-1}\|_\infty \end{align} Are there any upper bounds on the above normed term?
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38 views

An equivalent definition of the condition number of a matrix [on hold]

How can I prove that the condition number can't be expressed by $$\kappa(A)= \sup_{\lvert\lvert x \rvert \rvert=\lvert \lvert y \rvert \rvert} \lvert\lvert Ax\rvert \rvert/\lvert\lvert Ay\rvert ...
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21 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
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32 views

linear algebra (norm) [on hold]

Can someone explain to me the following definition - $\|T\|$ := $ \sup \{\|T(v)\| : v \in \mathbb{R}^n, \|v\| = 1\}$ where $T$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ and ...
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15 views

Does this property of time dependent sequences have a name?

For $i\in \mathbb{N}$ let $ \chi_i \colon \mathbb{R}^+ \to \mathbb{R}$ be such that for each $t\in \mathbb{R}^+$ we have $\chi_i(t) \in \mathcal{l}_2$ Suppose further that ...
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40 views

Does it matter if you use big $L$ or little $l$ when talking about $L$-norms?

I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to. I ...
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1answer
24 views

What are those variations of norms called?

Let $V$ be a vector space with a function $\|\cdot\|$ on it that satisfies all the axioms of norms except for scalability condition $\|\alpha \mathbf{x}\| = |\alpha| \|\mathbf{x}\|$ replaced with ...
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3answers
48 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
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65 views

What does ||u|| mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
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25 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| ...
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2answers
18 views

Matrix tensor indices

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$ Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} ...
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26 views

Problem in showing that a norm is a norm on one space, but not on another.

I have the following question from a past paper: "Consider the two sets, $$A=\{g\in C^1([0,1]):g(0)=g(1)=0\}$$ and, $$B=\{g\in C^1([0,1]):g'(0)=g'(1)=0\}$$ both subsets of the vector space ...
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62 views

Find a norm so that its closed unit ball is the area between $y=x^2-1$ and $y=1-x^2$

As the title specifies, I need to find an explicit formula for a norm $|||\cdot|||$ so that: $$B_{|||\cdot|||}=\{\mathbf{x} : ||| \mathbf{x}|||\le1 \}$$ where $\mathbf{x}=(x,y)\in\mathbb R^2$, is ...
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68 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
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0answers
39 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: ...
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1answer
73 views

Prove a theorem involving normed spaces and completeness.

Theorem: If $(X,||\cdot||)$ and $(X,||\cdot||')$ are homeomorphic, then $(X,||\cdot||)$ is complete if and only if $(X,||\cdot||')$ is complete. So my goal is to prove that since they are ...
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2answers
34 views

The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
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1answer
20 views

limit of vector when norm goes to infinity

Consider a $x \in \mathbb{R}^p$ and $a \in \mathbb{R}^p$ be a fixed vector. If $| \cdot|$ is the Euclidean norm, what can we say about: $\lim_{|x| \to \infty} \dfrac{a^T x}{|x|}$. My intuition says ...
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42 views

Bounds for inner product of $Ax$ and $x$

Reading a math text, I found, with no proof given, the following assertion. Suppose $A$ is a real $n \times n$ matrix, and suppose the real part of its spectrum lies between $a$ and $b$; i.e., the ...
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63 views

proximal operator of weighted L1 norm

I hope to solve this problem. $$\min \quad \left\| CX \right\|_{1} $$ $$ \text{s.t.}\quad AX=b, X >0 $$ where $C \in \mathbb{R}^{m \times m}$, $X \in \mathbb{R}^{m \times n}$, $A \in ...
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2answers
25 views

Disk in $\mathbb R^2$ with uniform norm

I am having a trouble understanding a definition. The points are in $\mathbb R^2$, and the author defines $\delta(p, r)$ to be an $l_\infty$ disk of radius $r$ centred at $p$. I just learned what the ...
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9 views

On equality of frobenius norms with constraints

For a given matrix A, \begin{equation} \begin{aligned} &&&\| Z \|^2_F = \| A \|^2_F\\ &&& Tr(Z'Z) = Tr(A'A)\\ & \text{subject to} & & Z \succeq 0\\ ...
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1answer
51 views

$\ell^p\!,$ for $p\neq2$, is not an inner product space. [duplicate]

Consider sequence spaces of the form, $$\ell^p=\Big\{x=\left(x_j\right)_{j=1}^\infty \mathrel{}\big|\mathrel{} \sum_{j=1}^\infty \left\lvert x_j \right\rvert ^p\lt\infty\Big\}$$ for $1\le ...
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21 views

Point about the theorem and proof of the inner product being a continuous function.

In moving to show that the inner product, $\langle\cdot,\cdot\rangle$ is a continuous function I have the following theorem in my notes (also on page 59 of "Linear Functional Analysis", Rynne and ...
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43 views

Is there any interesting relationship between a Hermitian matrix and its corresponding entrywise absolute?

In general, a Hermitian matrix can have complex off-diagonal terms. Given any Hermitian matrix $[A]_{n,m}$, I can construct another matrix $[\vert A\vert ]_{n,m} =\vert A_{n,m} \vert$. I would like to ...
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19 views

Can absolute scalability be 'relaxed' to an equivalent condition in the properties of a norm?

All norms on a vector space $V$ must satisfy for any $x\in V$ $$\Vert \alpha x \Vert = \vert \alpha \vert \Vert x \Vert $$ for any scalar $\alpha\in R$. However, I've been told that an equivalent ...
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58 views

differentiation of 1-norm of a vector

Assume you want to find the minimum of the following expression $\|x\|_1 + \alpha \|Ax-b\|_2$ where $x\in R^N$. So basically I want to calculate the derivative of $\|x\|_1$ so I could finally set ...
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Is there a name for this function with properties…

Let $V$ be a vector space over an algebraic structure $\mathbb{A}$, and suppose we have a binary operation $\star:V^2\to V$. Consider a function $f:V\to \mathbb{A}$ with the property that $$f(x\star ...
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37 views

Questions about the finite-dimensional normed space of polynomials of degree at most two.

Take $X:=P_2([0,1])$, the polynomials of degree at most $2$ over $[0,1]$ and consider the $2$-norm on this space. For any $x\in X$ we have that, $$\|x\|_2=\left(\sum_{i=1}^n|x_i|^2\right)^{1/2}$$ ...
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47 views

Possible ways to induce norm from inner product

Let $ S $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. Can this norm be induced from inner product only through $\lVert \cdot \rVert = ...
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In $\mathbb{R}^{n}$ the norms induced by inner products are equivalent.

I need your help to proceed in proving this theorem: Let $\;\left\|\cdot\right\|_1\;$ and $\;\left\|\cdot\right\|_2\;$ be norms on $\mathbb{R}^{n}$ induced by inner products. Then they are ...
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20 views

Proving that if $B^1_\delta(0) \subset B_1^2(0)$ and $B_\epsilon^2(0)\subset B_1^1(0)$ then the norms are equivalent

I want to prove the following proposition: Let $||\cdot||_1$ and $||\cdot||_2$ two norms on X, and suppose there exist $\delta, \epsilon>0$ such that $B^1_\delta(0) \subset B_1^2(0)$ and ...
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35 views

How does sketching norms show that they are equivalent?

I have the following statement in my notes: "You might want to check by drawing the sets of all $x\in\mathbb R^2$ such that $\|x\|_1=1$,$\|x\|_2=1$,$\|x\|_\infty=1$ that indeed these norms are ...
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1answer
23 views

Derivative of the squared $L^2$ norm of a vector function

Given the function: $$y({\bf w}) = \| \,{\bf w\, w^T x - x } \,\|^2_2$$ I'm trying to understand how to get the derivative $\frac{\partial y}{\partial\bf w}$, where $\bf w$ and $\bf x$ are vectors. ...
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2answers
77 views

Prove the the uniform, $L^1$ and $L^2$ norms are not equivalent.

Two norms $||\cdot||_1$ and $||\cdot||_2$ on a vector space are equivalent if there exist constants $c_1$, $c_2$ such that $$c_1||x||_1 \le ||x||_2 \le c_2||x||_1$$ for all $x\in X$ (I have proved ...
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74 views

Taking the limit $\lim_{p\rightarrow \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p$

Taking the limit $$\lim_{p\rightarrow \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p$$ First I think the expression after taking the limit will depend on the function $f$. In my attempt, ...
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45 views

Line segment in the unit sphere

I want to prove the following statement Let $X$ be a normed linear space, with linearly independent vectors $x,y$, such that $\|x\|=1=\|y\|$, with $\|x\|+\|y\|=\|x+y\|$, then there is a line ...
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47 views

Understanding convergence in normed spaces and the language used when talking about norms.

We have the following definition about convergence in a normed space: "Let $(x_n)_{n=1}^\infty$ be a sequence in a normed space $(X,\|\cdot\|)$. We say that $x_n\to x$ in $X$ if, $$d(x_n,x)\equiv ...
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24 views

Vector norm - Understanding the definition of the unit sphere

If $\|x\|=1$ just means the vector $x$ has length one - Then why is the unit sphere defined as $S=\{x\in X| \quad \|x\|=1\}$? let $X$ be a normed linear space with the Euclidean norm, then letting ...
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31 views

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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245 views

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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27 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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40 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 ...