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

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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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The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
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Distance between point and linear Space

Suppose $E$ is a normed vector space. Let $f$ be a continuous linear functional on $E$ and denote by $M$ the Kernel of $f$. Let $x\in E$. How to show that ...
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How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
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Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
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3answers
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How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
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2answers
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Understanding the properties and use of the Laplacian matrix (and its norm)

I am reading the wikipedia article on the Laplacian matrix: http://en.wikipedia.org/wiki/Laplacian_matrix I don't understand what is the particular use of it; having the diagonals as the degree and ...
3
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1answer
900 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 ...
9
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690 views

From norm to scalar product

In the Eculidean Space, one can automatically define a norm if a specific scalar product is given. On the contrary, it's not always true. A p-norm is a scalar product if and only if p=2. My question ...
2
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3answers
441 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
4
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1answer
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Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
3
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2answers
253 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
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2answers
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Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
8
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1answer
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Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
5
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Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
4
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Examples of metric spaces which are not normed linear spaces?

Give an example of a metric space which is not a normed linear space. Justify your example.
8
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For any three vectors $x,y,z\in\mathbb{R}^d$, we have $ \|y-z\|\cdot\|x\|\leq\|x-y\|\cdot\|z\|+\|z-x\|\cdot\|y\|$

Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$ \|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| ...
8
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2answers
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What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are ...
3
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3answers
939 views

equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
7
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Existence of norm for C*-Algebra

I was wondering wether one can always find/construct a norm which turns an involutive algebra into a C*-algebra. For sure, if it exists it is unique, but does it always exist. If not can you provide a ...
4
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2answers
858 views

I am not sure how to calculate this norm?

I have the following matrix: $$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ What is the norm of $A$? I ...
4
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1answer
247 views

Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, ...
3
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2answers
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How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
2
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1answer
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How can I give a bound on the $L^2$ norm of this function?

I came across this question in an old qualifying exam, but I am stumped on how to approach it: For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let $$(Vf)(x) = ...
0
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1answer
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Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j_=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
0
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1answer
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Proof of matrix norm property: submultiplicativity

I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define ...
0
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3answers
861 views

2-norm vs operator norm

I have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the ...
4
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2answers
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Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle ...
4
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1answer
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Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
4
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Is norm non-decreasing in each variable?

Let me try again. Suppose $\|\cdot\|$ is a norm in $\mathbb{R}^n$ and let $$f(x_1,...,x_n)=\|(x_1,...,x_n)\|$$ where $x_i\geq 0, \forall i$. I want to prove or disprove that $f$ is an nondecreasing ...
2
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2answers
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Norms Abstract Analysis

I have a question relating to norms and have been giving functions and need to state whether they are norms or not... which of the following are norms on $\mathbb{R}^2$? Give reasons for your ...
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Subordinate matrix norm

I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
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1answer
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Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
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1answer
191 views

A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm ...
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1answer
421 views

Proving the triangle inequality for the L-2 norm $||x||_2 = \sqrt{x_1^2+x_2^2\ldots+x_n^2}$

I want to prove the triangle inequality for the l2-norm $||x||_2$: $$||x||_2 = \sqrt{x_1^2+x_2^2+\ldots+x_n^2}$$ $$\begin{align} \sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} + {y_i}} \right)}^2}} ...
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0answers
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Distance metrics with kmeans

Context: I'm trying to derive some formulas for computing the "mean" in the K-means algorithm. So given an assignment of $m$ data points to $k$ clusters, find a formula to recompute the mean of the ...
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Use norm on $\mathbb{R}$^k to prove $\mathbb{R}$^k is complete [on hold]

got it. No more help needed. Thanks.
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1answer
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Contraction map in extending domain from Dense subset to $L^{2}$

This thread is about extending a dense domain $D \subset L^{2}$ into $L^{2}$. I do not understand what Deyton means in his comment about getting contraction map when doing this. I cannot see any ...
16
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1answer
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On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
14
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1answer
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Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
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Maximum subset sum of $d$-dimensional vectors

This is a $d$-dimensional generalisation of the post Inequality with Complex Numbers. (See my comment under Robert Israel's answer.) Generalising Potato's proof for $d$-dimensions, we can show the ...
5
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1answer
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Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
6
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1answer
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Is there a geometric meaning of the Frobenius norm?

I have a positive definite matrix $A$. I am going to choose its Frobenius norm $\|A\|_F^2$ as a cost function and then minimize $\|A\|_F^2$. But I think I need to find a reason to convince people it ...
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0answers
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Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
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2answers
543 views

Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$

I am studying matrix norms. I have read that $\|A\|_{\infty}$ is the largest row sum of absolute value and $\|A\|_{1}$ is the highest column sum of absolute values of the matrix $A$. However, I am ...
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Monotone matrix norms

[Ciarlet 2.2-10] Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if ...
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Equivalent norms

Whenever two norms are equivalent in the sense that $||x||_1\le c_1\cdot ||x||_2$ and $||x||_2\le c_2\cdot ||x||_1$, they generate the same topology. Is the reverse also true, i.e. if a topology is ...
7
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1answer
617 views

On matrix norm equivalence

For finite dimensional spaces, all norms are equivalent, i.e. there exist constants say $A,B$ such that for all matrices from the $\mathbf M \in R^{d\times d}$ (let $d$ be a fixed positive integer) ...
5
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602 views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
4
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1answer
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Geometric interpretations of $||z||_p = 1$?

Here $z = a + bi$, with $a, b, \in \mathbb{R}$ and $||z||_p = \sqrt[p]{|a|^p + |b|^p}$. With $p = 1$, this is just diamond (square rotated 45 degrees) of side=$\sqrt2$ centered at the origin. With ...