Tagged Questions

1
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0answers
21 views

Does convex and radially open imply open?

I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace. Here the 'openness' we are talking about is from any normed space. ...
6
votes
2answers
65 views

Coercive bilinear form on Hilbert space

I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. Consider a continuous symmetric bilinear form $B$ on a ...
3
votes
1answer
79 views

Skew symmetric matrix decomposes

I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $ R $ such that $ R^T A R = M$, ...
2
votes
3answers
30 views

difference between normed linear space and inner product space

I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other ...
0
votes
2answers
42 views

Self adjoint positive definite and product of two operators

I just wanted to ask you whether a theorem that I have found on wikipedia is correct! I have found a theorem that says, that a matrix is positive definite if and only if it is equal to the product of ...
0
votes
1answer
62 views

$W^0$ is a subspace of $V^*$

If $W\subset V$ is a subspace of $V$, and $W^0=\{f\in V^* | f(v)=0~\forall~v\in W\}$, then how do I show that $W^0$ is a subspace of $V^*$?
0
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0answers
26 views

Normal endomorphism

I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
0
votes
2answers
47 views

Self adjoint operator

I am looking in the space of test functions $ \{f \in C^\infty|f^{(n)}(a)=f^{(n)}(b)=0\};n \in \mathbb{N}_0\} $whether the n-th derivative is a self adjoint operator. the dot product is given by ...
0
votes
1answer
67 views

Representation of a bilinear form on an Hilbert space

Given a bilinear symmetric form $b(u,v)$ on a Hilbert space. I need to know some very basic facts. A reference where these are discussed would be greatly appreciated. 1) There exists a symmetric ...
1
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0answers
32 views

Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
2
votes
1answer
51 views

Second annihilator of subspace is the subspace itself?

Let $X$ be a Banach space over $\mathbb{C}$ with dual space $X'$ and let $M,N$ be subspaces of $X',X$ respectively. Define the annihilator subspaces of $M$ and $N$ as $$ M_\circ = \{x \in X: f(x) = ...
2
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0answers
23 views

what to do if it's not direct sum?

Suppose $X=Y+Z$ is Banach, $Y$ and $Z$ are closed subspaces. I want to show there exists $\alpha>0$ such that $\forall x \in X, \exists$ $y \in Y$ and $z \in Z$ such that $x=y+z$ and $\|y\|+\|z\| ...
1
vote
1answer
45 views

Is this set $\{ p(x): x\in \operatorname{bco} A\}$ bounded in $\mathbb{R}$?

$\newcommand{\bco}{\operatorname{bco}}$Here are some terminologies. Definition. Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, denoted $\bco A$, is the ...
1
vote
1answer
40 views

About Balanced-Convex Hull of a Set

Definition. $\newcommand{\bco}{\operatorname{bco}}$Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, deonoted $\bco A$, is the intersection of all balanced-convex ...
2
votes
2answers
50 views

little question about linear operators

Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
4
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0answers
47 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operator $E ...
1
vote
1answer
33 views

some inclusions regarding linear operators

Let $H$ be a Hilbert Space and $T:H\rightarrow H$ a linear operator. Let $T^*$ be the adjoint operator of $T$ and let $\operatorname{Cl}(X)$ be the topological closure of the set X and $X^{\perp}$ ...
4
votes
1answer
96 views

Name for this simple inequality

Let $x,y$ vectors in $\mathbb{R}^3$. From $$\Vert x+y\Vert^2\geq 0$$ it follows that $$2x\cdot y\geq -\Vert x\Vert^2-\Vert y\Vert^2$$ Has this inequality a name?
1
vote
1answer
32 views

linear independence in the dual space

If $V$ is a $N$ dimensional vector space and $l_1,....,l_k$ are linearly independent elements of $V^*$. How to prove that map $V\to R^k$ given by $v\to (l_1(v),...,l_k(v))$ is surjective? It does not ...
1
vote
1answer
36 views

quasi-inner product problem

Let $X$ a vector normed space on $\mathbb{R}$ and $a:X\times X\rightarrow\mathbb{R}$ such that $a(x,x)\ \geq\ 0,\;\;\; \forall\ x\in X$. $a(x,y)\ =\ {a}(y,x),\;\;\; \forall\ x,\ y\in X$. $a(\alpha x ...
7
votes
1answer
60 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
1
vote
1answer
68 views

Separable spaces

Suppose we have a closed separable subspace $A$ of a non separable Hilbert space $H$, then is $A^\perp$ separable. If so what would be a way to prove it, if not why.
2
votes
1answer
34 views

dot product and relationship to cosine of its angle

In our class, we defined that the term $$ \cos(\alpha)=\frac{\langle x,y \rangle}{\sqrt{\langle x,x\rangle \langle y,y \rangle}}$$ is equal to the cosine of the angle enclosed by $x$ and $y$. ...
0
votes
1answer
70 views

How to sample uniformly from an $\epsilon$ ball?

Given a real rectangular matrix $X$, I would like to uniformly sample from the set of real rectangular matrices $\mathbb{M}$ that satisfy $||X-S||\leq \epsilon, \forall S\in\mathbb{M}$ and for a fixed ...
1
vote
1answer
62 views

Orthogonality, proof in Linear Algebra/Functional Analysi

Problem: Let $M$ be a complete subspace Y and $x \in X$ fixed. Then $z = x - y \perp Y$ Part of the Proof If $z \perp Y$ were false, there would be a $y_1 \in Y$ such that $\langle z, y_1 ...
0
votes
1answer
22 views

Example of a non-separating family of seminorms

Let $X$ be a real vector space and let $P$ be a family of seminorms on $X$. We say that $P$ is separating if for $x\in X$ with $x\neq 0$ we can find $p\in P$ such that $p(x)\neq 0$. I got no problem ...
2
votes
2answers
26 views

Function space of a finite set and $\Bbb R^n$

I read in a tutorial that a function space $F(S, \mathbb{R})$ of a finite set $S$ of cardinality $n$ has dimension $n$. To be clear $F(S, \mathbb{R})$ is the set of all functions defined on the set ...
0
votes
0answers
37 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
0
votes
1answer
55 views

Is it always true that the inner product is a map from a vector space to a scalar field?

In other words, is the inner product a bilinear functional? That is for $x,y \in V$ where $V$ is a vector space, is it always true that $\langle . \rangle: V\to \mathbb{F}$. Is it ever possible that ...
0
votes
1answer
26 views

Symmetric Operator with Different dot products

If I have a symmetric operator $A$ in a metric space $\mathscr{M}$. Then $\langle Au,v\rangle =\langle v,Au\rangle $ with the dot product defined in $\mathscr{M}$. My question is, if I keep the same ...
0
votes
1answer
27 views

Inner product and norm of a function

I have recently started a undergrad. linear algebra course in which these definitions came up: Let $V$ be the vector space $C[a,b]$ of all continuous functions on $[a,b]$. Then the inner product and ...
1
vote
0answers
102 views

Kernel of Fractional Differential Operator

Suppose we have a fractional differential equation: $$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]y(t)=0$$ where $\nu=\frac{1}{q}$ and $q\in\mathbb{N}$ and y is an analytic ...
1
vote
2answers
60 views

orthogonal complement of symmetric matrices

How do I can prove that the orthogonal complement of space of symmetric matrices is the space of skew-symmetric matrices? With the inner product $\langle A,B\rangle = \mbox{tr}(A^TB)$. Thanks in ...
2
votes
1answer
41 views

Self-adjoint operator in the space of twice continuously differentiable functions

There is a problem in the textbook with which I am having difficulties. Prove that operator $A$: $Ay=xy''+y'$ defined on space of twice continuously differentiable functions (scalar product is ...
2
votes
1answer
92 views

Relation between positive definite Hermitian matrices with their inverses

Let $A$ and $B$ be two positive definite Hermitian matrices. Show that the Hermitian matrix $$C\ =\ A^{-1} + B^{-1} - 4(A + B)^{-1}$$ is also positive definite. Thanks in advance.
1
vote
0answers
35 views

Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
8
votes
1answer
190 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
4
votes
1answer
80 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
2
votes
3answers
56 views

Solving for positive semidefiniteness

Given a real matrix M, is there a matrix function f(M) such that $f(M)-M$ is guaranteed to be positive semidefinite, other than the idea of multiplying $M$ with its transpose and apart from the ...
1
vote
1answer
40 views

Generalization of zero-diagonal square matrices to linear operators

Which linear operators in Banach or Hilbert spaces (e.g., partial differential operators or some other operators in functional spaces) are generalizations of square matrices $A=(a_{ij})$ such that ...
2
votes
3answers
68 views

norm of a linear operator

On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator. Is this a theorem of some sort? If so, how can it be ...
1
vote
1answer
65 views

$l^{p}$ is not finite dimensional

Well, the exercise was to prove that $l^{p}$ is not finite dimensional space for $p$=2. I did it proving that the unit ball is not compact. Easy. However, i was trying to build an element $x \in ...
1
vote
2answers
60 views

Why the nontrivial nullspace of a linear has codimension 1?

The nullspace of a linear functional that is not $\equiv 0$ is a linear subspace of codimension $1$. I don't understand this statement on page 57, Functional Analysis(Pater Lax). Does it mean the ...
1
vote
1answer
31 views

Linear functionals on space of all converges sequences.

Suppose that $V$ is the space of all converges real sequences, $(x_i)_{i=0}^{\infty}$ How to show that every absolute converges series, $\sum_{i=0}^{\infty}a_i$ defining a linear functional with: ...
2
votes
1answer
134 views

Reproducing Kernel Hilbert Spaces for Dummies

I am in the middle of some machine learning paper that states that for function $f$, imposing the norm constraint, $\|f \|=1$, corresponds to an orthogonal projection onto the direction selected in ...
2
votes
1answer
52 views

Families of vectors in finite dimensional Hilbert space

This is an exercise left for the reader in proof of Proposition 1.12 in Pisier's book "Introduction to Operator Space Theory". Assume that we have $n$ vectors $x_1, \dots, x_n$ in $\ell_2^N$ (where ...
1
vote
0answers
24 views

pseudoinverse under change of norm

Let $X$ be a Hilbert space. Let $T : X \rightarrow X$ be a linear mapping. Suppose we have two scalar products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ on $X$. Let $T_1$ and ...
1
vote
0answers
68 views

Relationship between eigenvalues of a matrix and its derivative

Is there any relationship between the derivative of a matrix which depend by a parameter and its eigenvalues?
0
votes
0answers
21 views

Study of the invertibility of a matrix [duplicate]

I have to study the invertibility of this matrix $$A(k)=\bigg[\bigg(\alpha_j-\frac{ik}{4\pi}\bigg)\delta_{jj'}-\tilde{f}(y_j-y_{j'})\bigg]_{jj'}$$ where $\tilde{f}(x)$ is ...
2
votes
0answers
51 views

Invertibility of a matrix

I have to study the invertibility of this matrix $$A(k)=\bigg[\bigg(\alpha_j-\frac{ik}{4\pi}\bigg)\delta_{jj'}-\tilde{f}(y_j-y_{j'})\bigg]_{jj'}$$ where $\tilde{f}(x)$ is ...

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