All Questions
Tagged with banach-spaces linear-algebra
38 questions
1
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1
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151
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Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 113
4
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0
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108
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Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
- 153
0
votes
1
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317
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A variation of the Riesz Lemma
Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
- 153
1
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1
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179
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Definition and properties of tangent functional
I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$.
We let $\tau(x, \cdot)$ denote the ...
- 297
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1
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446
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Parallelogram law for vectors of equal length
Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...
- 1,361
1
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1
answer
155
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Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
- 1,054
2
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1
answer
118
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Commutation of linear maps and extreme points
Let $X,Y$ be real vector spaces, $T: X\to Y$ be a linear map, and fix a nonempty $S\subseteq X$ (we do not assume that $S$ is neither convex nor compact (indeed, right now we do not assume any ...
- 1,511
2
votes
1
answer
133
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Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
- 5,405
0
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0
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45
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Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
5
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471
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A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?
In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
- 597
0
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1
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126
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Tauberian operators
Let $X$ be a Banach space non reflexive and $T$ from $l_2(X)$ to $l_2(X)$ a bounded operator defined by:
$$T(x_n )=\frac{x_n }{n}.$$
We know that :
$$T^{**-1}(l_2(X))=\{x_n^{**} \in l_2(X^{**}) : \...
4
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1
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174
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A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,529
6
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1
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203
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How to calculate the volume of a section of a convex body?
The following is essentially a partial case for my previous question.
Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$....
- 5,529
5
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0
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350
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How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
17
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2
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513
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On a special type of normed linear spaces
Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying
$$
\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}
$$ is a group ...
- 1,209
-1
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1
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132
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About a property in a reflexive Banach space
Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
- 2,118
13
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2
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653
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The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 536
9
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2
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338
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Does $End(V)$ remember $V$, where $V$ is a locally convex space?
Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a ...
- 43.2k
3
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422
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Isometries between subspaces of finite-dimensional vector spaces
I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.
Taking $n \le m$, one ...
- 31
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3
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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 ...
- 391
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263
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Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 305
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125
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Estimating $\ell^p$ and $\ell^q$ norms on a convex cone
For $1 \le p \le q \le \infty$, I need an inequality bounding the $\ell_q$ norm from above by the $\ell_p$ norm on $\mathbb{R}^n$: finding a $\lambda$ so that
$$
\Vert v \Vert_q \le \lambda \Vert v \...
- 10.1k
1
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1
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158
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When do we have $B_Y\subset T(B_X)$ if and only if $\overline{B_Y}\subset T(\overline{B_X})$?
Let $X$,$Y$ be normed spaces, $T:X\to Y$ be a bounded linear operator. Denote the open and closed unit balls by
$$
B_X:=\{ x\in X\ |\ \|x\|<1\} \\
\overline{B_X}:=\{ x\in X\ |\ \|x\|\le1\}
$$
and ...
- 1,245
1
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0
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76
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Linear independence of an odd set of measurable functions
Let $g(t)$ be a convex positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}...
- 5,405
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487
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What is the trace of this operator in $L^\infty$ (if this question make sense)?
Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator :
\begin{array}{...
- 283
8
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0
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421
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Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
- 6,206
6
votes
2
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405
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$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$
Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le \...
- 23k
1
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1
answer
2k
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Constructing a continuous matrix valued function
Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
- 133
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0
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648
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Closed-form expressions for dual norms of real normed vector spaces
Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\...
- 4,897
2
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1
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386
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Decomposing bilinear forms in Hilbert spaces
You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
- 1,211
3
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2
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2k
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How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?
Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V_s$...
- 4,897
13
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1
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732
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What is the "positive part" of the unit ball in $M_n(R)$ ?
In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...
- 52.3k
7
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1
answer
530
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Do real vectors attain matrix norms?
I apologize if the following question ends up being too elementary for this website; I asked it on math.SE a week ago and it remains unanswered.
Let $A$ be an $n \times n$ matrix with real entries ...
- 571
5
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1
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528
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Completely bounded maps on Mn
The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property:
$\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and $||\phi_n||_{cb}\...
- 4,705
2
votes
2
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308
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Analogue of an orthogonal subspace in a noneuclidian normed space
This question is related to https://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question.
...
- 3,595
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Is exp(rA) = (exp(A))^r for real r and A in a Banach space?
Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
...
- 657
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1
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2k
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Banach-Mazur distance between $\ell^p$-norms
Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then
$$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$
is an operator norm ...
- 52.3k
6
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2
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2k
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Inner products and norms
Let $f:[n]\times [n] \rightarrow [0,1]$ be a function from pair of integers to the real interval [0.1]. I would like to find sets of complex vectors
$X= \{x_i\}$ and $Y=\{y_j\}$ satisfying $x_i\cdot ...
