Questions tagged [norms]

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34 votes
5 answers
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Linearity of the inner product using the parallelogram law

A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula: $2\langle u,v\rangle ...
Andrew Stacey's user avatar
31 votes
0 answers
2k views

Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
Mikhail Ostrovskii's user avatar
23 votes
3 answers
15k views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
Paglia's user avatar
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22 votes
6 answers
1k views

Under what conditions $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$ implies isometry?

Let $X, Y$ be normed space and $f:X\to Y$ be a mapping. Assume that for all $n\in\mathbf{N}$, $$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$ Under what conditions this map will be an isometry? Thanks
user62498's user avatar
  • 813
18 votes
3 answers
5k views

What are the matrices preserving the $\ell^1$-norm?

So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
D. Rusin's user avatar
  • 381
18 votes
3 answers
931 views

Bounding the commutator [A,B] in terms of the numerical radius

Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that $$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$ The answer is ...
Denis Serre's user avatar
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18 votes
2 answers
1k views

An equivalence relation for norms

Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that $$ \frac{1}{\lambda} \left( \|x\|_1 +...
alvarezpaiva's user avatar
  • 13.2k
18 votes
1 answer
948 views

Extremal points of the unit ball of M_n(R) ...

The unit ball of ${\bf M}_n(\mathbb R)$ is a compact convex subset. As such, it is (Krein-Milman theorem) the convex envelop of its extremal points. So far, so good; but the unit ball depends of the ...
Denis Serre's user avatar
  • 51.5k
17 votes
2 answers
506 views

On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $$ \|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z} $$ is a group ...
user521337's user avatar
  • 1,189
17 votes
0 answers
220 views

GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to four given spheres, an Apollonian problem in $\mathbb{R}^3$. The center of that fifth sphere is one of the $16$ possible solutions to ...
Joseph O'Rourke's user avatar
16 votes
3 answers
690 views

An inequality for two independent identically distributed random vectors in a normed space

Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$? Some background information on ...
Iosif Pinelis's user avatar
16 votes
1 answer
3k views

A property that forces the NORM to be induced by an INNER PRODUCT

Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$, $$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$ I want to show that the norm is induced by an inner product. Any ...
user avatar
16 votes
1 answer
1k views

How many values determine a norm?

It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely. How many values do we need to specify in order to ...
Asaf Shachar's user avatar
  • 6,611
15 votes
2 answers
2k views

In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is a cross-posted on MSE here. Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
Nikhil Sahoo's user avatar
  • 1,175
15 votes
2 answers
1k views

Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$

The setup is as in this question: Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that $$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}...
Wolfgang's user avatar
  • 13.2k
15 votes
2 answers
888 views

Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?

I was reading the proof of Ostrowski's theorem, with an eye toward the Zariski-Riemann space (as well as adic space, Berkovich space, etc.) In the proof, the value group is always assumed to be in $\...
zxx's user avatar
  • 183
14 votes
1 answer
835 views

What are the applications of the Mazur-Ulam Theorem?

Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
Pietro Majer's user avatar
  • 56.5k
13 votes
4 answers
786 views

Are norms intrinsically $\mathbb{R}$-valued?

Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$? I have searched a ...
Jacques Carette's user avatar
12 votes
2 answers
924 views

which norms can be realized as operator norms?

Assume $(V,∥∥_V),(W,∥∥_W)$ are both finite dimensional normed spaces. We have the induced operator norm on ${\rm Hom}(V,W)$. It turns out that the operator norm is induced by an inner product iff ...
Asaf Shachar's user avatar
  • 6,611
12 votes
1 answer
441 views

Subtracting the weak limit reduces the norm in the limit

Question Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that $$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$ ...
Willie Wong's user avatar
  • 37.4k
12 votes
3 answers
364 views

Probability of $\ell_1$-norms of vertices of the rotated Hamming cube

Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...
arriopolis's user avatar
11 votes
2 answers
984 views

A "quadratic" triangular inequality

In a Euclidian space (Hermitian as well), say $\ell^2_n$, the following inequality holds true $$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2_n.$$ In other words, ...
Denis Serre's user avatar
  • 51.5k
10 votes
3 answers
1k views

Pathological product space norm

Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||_{X \...
Zuza's user avatar
  • 202
10 votes
1 answer
2k views

Counting norms on an infinite dimensional vector space

It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology). Is it known what happens when E ...
dionysos's user avatar
  • 101
10 votes
2 answers
5k views

Nuclear norm as minimum of Frobenius norm product

Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix. It is claimed that $$ \|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
Hans's user avatar
  • 2,169
10 votes
1 answer
1k views

On bi-invariant metrics on groups

I'm looking for a quick, snap-your-fingers proof of the following result: A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm. To be clear about the ...
alvarezpaiva's user avatar
  • 13.2k
10 votes
0 answers
253 views

Integral points on elliptic curve and the Lee norm

This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE: Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$. The ...
user avatar
9 votes
2 answers
802 views

What norms can be "universally" defined on any real vector space with a fixed basis?

Let $V$ be a real vector space and let $B = (b_\lambda)_{\lambda \in \Lambda}$ be a basis. So every $v \in V$ can be written uniquely as a linear combination $$ v = c_{\lambda_1} b_{\lambda_1} + c_{\...
AlpinistKitten's user avatar
9 votes
2 answers
490 views

Projections onto $n$-codimensional subspaces of a Banach space: norms.

Hello, I'd like some help to find an answer I've been looking for since this morning. Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...
LaTortoise's user avatar
9 votes
1 answer
640 views

Banach space with uncountable basis

We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach ...
Anupam's user avatar
  • 479
9 votes
1 answer
468 views

Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$

I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows: As a set $D=E\oplus uE \oplus u^2 E$ where $u$...
user300's user avatar
  • 215
9 votes
1 answer
494 views

Regular $p$-norm of a matrix

Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as ...
Jochen Glueck's user avatar
9 votes
1 answer
1k views

Under what conditions a linear automorphism is an isometry of some norm?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and $T: V \to V$ is a (linear) isomorphism. When is it possible to construct a norm on $V$ making $T$ an isometry? (Hopefully,...
Asaf Shachar's user avatar
  • 6,611
9 votes
1 answer
273 views

Continuously varying norms

Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number $$ d =...
Jakob Werner's user avatar
  • 1,093
8 votes
3 answers
1k views

Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...
Don John Prep's user avatar
8 votes
3 answers
2k views

Bounding supremum norm of Lipschitz function by L1 norm

Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ? In ...
Aurelien's user avatar
  • 191
8 votes
2 answers
496 views

Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$. There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...
Rubi Shnol's user avatar
8 votes
2 answers
876 views

Sum of the norm of polynomials

Let $\bar D$ denote the closed unit disc in the complex plane. Consider the function $f:\bar D\longrightarrow \mathbb{C}$, defined as $f(z)=z$ for all $z\in \bar D$. Let $n\in \mathbb{N}$. For $1\leq ...
User93709's user avatar
  • 355
8 votes
1 answer
347 views

Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality $$ |\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} \|...
Transcendental's user avatar
8 votes
0 answers
719 views

Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
Piotr Migdal's user avatar
  • 1,592
7 votes
3 answers
897 views

Norms as Points in $C(X)$

$\newcommand\abs[1]{\lvert{#1}\rvert}$Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$. For each point $x$, there is a ...
Ronald J. Zallman's user avatar
7 votes
2 answers
611 views

If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?

Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class. Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
Frederik Ravn Klausen's user avatar
7 votes
1 answer
1k views

Operator norm of square root of matrix vs original

If I have a nonsymmetric matrix whose operator norm is $\leq 1$ and square root it, does its operator norm remain below $1$? More formally, I want to know whether there is always at least one square ...
user149538's user avatar
7 votes
1 answer
2k views

Equivalence of entrywise 1-norm and Schatten-1 norm

Let $A \in \mathbb{R}^{m\times n}$ and $\|A\| = \sum_{i, j} |A_{i,j}|$. I am looking for constants $\alpha, \beta \in \mathbb{R}$ such that $\alpha \|A\| \leq \|A\|_* \leq \beta \|A\|$ The function ...
rodms's user avatar
  • 399
7 votes
2 answers
509 views

Proving an infinite norm minimization problem has finite support (non-convex p-norms)

Consider an optimization problem over infinite variables: $$ \begin{align} \min_{x}~& {\left\lVert{x}\right\rVert }_p \\ \text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\...
Itay's user avatar
  • 661
7 votes
1 answer
510 views

Real vs complex norm for polynomials

Let $d \in \mathbb{N}$. We denote by $C_d$ the best (smallest) constant satisfying that $$ \sup{\{ |P(z)| \colon z \in \mathbb{D} \}} \leq C_{d} \, \sup{\{ |P(x)| \colon x \in [-1,1] \}} $$ for every ...
TonyP's user avatar
  • 73
7 votes
1 answer
658 views

Norms of B-spline coefficients

In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar ...
Housen's user avatar
  • 176
7 votes
1 answer
323 views

Absolute value on tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$. Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{1}...
user223794's user avatar
6 votes
1 answer
978 views

Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?

Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
Peter's user avatar
  • 131
6 votes
1 answer
3k views

When do 0-preserving isometries have to be linear?

Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$. Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ . What ...
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