All Questions
Tagged with fa.functional-analysis linear-algebra
23 questions
51
votes
2
answers
5k
views
A strengthening of the Cauchy-Schwarz inequality
Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
- 6,351
35
votes
2
answers
2k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
25
votes
16
answers
4k
views
functions satisfying "one-one iff onto"
Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...
13
votes
1
answer
1k
views
A generalization of the Powers-Stormer inequality
The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...
- 2,019
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
8
votes
1
answer
386
views
Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$
Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$.
Question. Can we prove this?
$$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
- 519
8
votes
1
answer
1k
views
Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
- 81
8
votes
1
answer
485
views
An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...
- 563
8
votes
3
answers
691
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
- 12.5k
7
votes
3
answers
3k
views
Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space
I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask.
First, consider the following form ...
- 29.8k
6
votes
1
answer
236
views
Potential p-norm on tuples of operators
Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}.
$$
Q: Is this a norm?
...
- 3,984
5
votes
1
answer
332
views
On a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
5
votes
0
answers
350
views
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
5
votes
3
answers
1k
views
Constant rank theorem for Banach spaces
Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
- 2,099
4
votes
1
answer
174
views
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
3
votes
1
answer
773
views
Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?
Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.
But do we have any quantitative ...
- 1,893
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
0
answers
136
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
1
vote
1
answer
223
views
A linear algebraic q-difference equation [SOLVED]
I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...
- 133
1
vote
1
answer
322
views
A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
- 4,058
1
vote
1
answer
114
views
Is a $1_A \otimes U$ invariant subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ a product $V_A \otimes \mathcal{H}_B$?
Consider a subspace $V$ of $\mathcal{H}_A \otimes \mathcal{H}_B$, with $\mathcal{H}_A$ and $\mathcal{H}_B$ finite-dimensional Hilbert spaces, that is $1_A \otimes U$ invariant for all unitary ...
- 133