1
vote
0answers
17 views

Usual linear combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
0
votes
0answers
22 views

Canonical term for $\overline X / X$ where $X$ is a normed space.

Let $X$ be a normed vector space. Let $\overline X$ denote its completion. Is there a canonical name for the quotient space $\overline X / X$? Some authors seem to use "torsion" as a name, but I ...
0
votes
1answer
53 views

Why $\langle a,x\rangle = \langle b,x\rangle,\forall x\in X\implies a=b$ [closed]

Let $X$ be (possibly infinite-dimensional) Hilbert space. How can we show that if $$\langle a,x\rangle = \langle b,x\rangle,\forall x\in X$$ then $a=b$?
0
votes
0answers
20 views

Proving that integrator operator of a kernel satisfies a specific peroperty

I am trying to prove that a integrator operator of a kernel satisfy a specific property say $\phi$. By integrator operator for non-negative definite kernel $\mathcal{K}$ I mean $T_{\mathcal{K}}$ such ...
1
vote
0answers
39 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
0
votes
0answers
117 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
1
vote
3answers
37 views

$(Ax,x)>0 \forall x$ implies $A$ selfadjoint?

I read in Reed & Simons's Functional Analysis (Vol.1, pg. 194) that a positive operator, i.e, an operator $A$ such that $(Ax,x)>0 \forall x$ on a complex Hilbert space is selfadjoint, but this ...
6
votes
1answer
92 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
0
votes
1answer
28 views

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable. It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
5
votes
0answers
100 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
0
votes
1answer
17 views

Analog of eigenvalue bound for a general bounded operator

It's known that for a matrix, $\max|λ|≤\sqrt{tr(A^*A)}=\sqrt{∑_{i,j=1}^n|A_{i,j}|^2}$ where $\lambda$ denotes its eigenvalue. I'm wondering whether there's an analog of this inequality for a general ...
0
votes
1answer
44 views

comparing largest eigenvalue of two positive matrices

I have a conjecture that for any two positive matrices(all elements are positive, nothing about positive definite or symmetry) $A$ and $B$, if $A_{ij} A_{ji}>B_{ij} B_{ji}$, while there is certain ...
8
votes
2answers
119 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
0
votes
1answer
26 views

Show that $(\overline{\text{lin} A})^{\perp}=A^{\perp}$

I have to show that $(\overline{\text{span} A})^{\perp}=A^{\perp}$ where $A$ is a subset of a Hilbert space. Any idea how to do it?
2
votes
0answers
106 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
0
votes
1answer
22 views

Vector minimizing the norm

Let $X_0$ be a finite-dimensional linear subspace of the linear normed space $X$. Show, that for every vector $x \in X$ there exists at least one vector $x_0 \in X_0$ such that: ...
1
vote
1answer
37 views

Revisiting the product rule for derivatives

Let $E=C^{\infty}(\mathbb R, \mathbb R)$ Consider a linear transformation on $E$: $\delta$ such that $\forall f, g \in E, \delta(fg) =g\delta(f) +f\delta(g)$ Prove that there is some ...
0
votes
1answer
33 views

Example of bijection operator

Is possible to find an example of a Banach space $E_1$ and normed vector space $E_2$ and bijection operator $A \in \mathcal{L} (E_1, E_2)$, that $A^{-1}$ will not be bounded?
4
votes
2answers
399 views

Matrices A+B=AB implies A commutes with B

$A$ and $B$ are $n\times n$ matrices and $A+B=AB$. I have an interesting proof that this implies $A$ commutes with $B$, but the proof only works when $||B|| \lt 1$. I'm looking for a way to salvage ...
1
vote
0answers
28 views

Characterizing direct sums

Let $U,V$ be vector spaces. Let $T: U \to V$ be a linear map. The codimension of $T$ is defined to be $\mathrm{dim}(V) - \mathrm{dim}(\mathrm{im}(T))$. My questions are: (1) given the subspace ...
3
votes
1answer
35 views

Similarity transformation of a linear operator

I've seen in some books that given a differential operator $$\frac{d}{dx}$$ under a similarity transformation we get $$\frac{d}{dx}\rightarrow ...
0
votes
1answer
42 views

Some infinite dimensional linear algebra, kernels of linear maps

I'm studying functional analysis (namely weak convergence) and need to prove the following result: if $f,f_1,\ldots f_n$ are some linear maps $X\to \mathbb{C}$, where $X$ is a vector space over ...
0
votes
1answer
68 views

Equivalence of norms in finite-dimensional spaces

Assume we have a separable, reflexive Banach space $X$ such that $\{e_{i}\}$ is a basis. Let $X_{n} = \text{span}\{e_{1},\ldots,e_{n}\}$ be finite-dimensional subspaces where we define members ...
5
votes
0answers
68 views

Inner product on $C(\mathbb R)$

With Axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. Of course, the space would be not complete under the norm induced by the inner product. My question is, is it ...
2
votes
4answers
86 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
1
vote
1answer
35 views

Transpose of a differential operator

Let $H$ be a diagonalizable matrix (not necessarily Hermitian). Then, it induces a biorthogonal left and right vectors, such that $$ ...
0
votes
1answer
27 views

Set of polynomials

I want clarification on the following question: Let $\{c_0,c_1,c_2,\dots,c_n\}$ denote a set of $n+1$ distinct elements in $\mathbb{R}$. Define the set of $n+1$ polynomials. $$f_j(x)=\prod_{k=0,k\ne ...
0
votes
1answer
40 views

If I know $\langle g(f), a_i \rangle$ for all $i$ where $a_i$ is a basis, do I know the coefficients of $f$?

Here I work on a bounded domain. Let $A=\text{span}(a_1, ..., a_n)$ where $\{a_i\}_{n=1}^\infty$ is a basis (not orthonormal) of $L^p \cap L^2$. Suppose $f \in A$. Let $g:\mathbb{R} \to \mathbb{R}$ ...
2
votes
3answers
34 views

Given a basis $a_i$ to a space, does knowing $\langle f, a_i \rangle$ for each $i$ determine $f$?

Let $A=\text{span}(a_1, ..., a_n)$ where $a_i$ is a basis (not orthonormal) wrt inner product $\langle ,\rangle$. Suppose $f \in A$. If I know $\langle f, a_i \rangle$ for each $i$, does that ...
0
votes
2answers
32 views

orthogonal projection onto orthogonal complement

If $V=M \oplus M^{\perp}$. For any $v\in V$, the orthogonal projection of $v$ onto $M$ along $M^{\perp}$ is well defined. Can we take the orthogonal projection of $v$ onto $M^{\perp}$ along $M$?
0
votes
1answer
13 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
4
votes
0answers
97 views

Do infinite dimension vectors hold the same properties as finite dimension vectors?

I am learning about vectors and am wondering if finite dimension vector operations such as the dot product hold for infinite dimensional vectors?
-1
votes
1answer
35 views

Writing a linear functional as a linear combination

Let $X$ be a NLS and $f,f_1,\ldots,f_n$ be linear functionals on $X$. Let $\bigcap_{i=1}^n\ker f_i \subseteq \ker f$. Show that $f=\sum_{i=1}^n a_i f_i$ for some scalars $a_i$.
0
votes
0answers
51 views

Complete Normed Space => Uncountable Hamel basis not by Baire

I need to show that a complete normed space X has no countable Hamel basis. One possibility is to with Baire's theorem. I, however, try to give an explicit sequence, namely: For a contradition, let ...
0
votes
0answers
22 views

Cannot use alternative definition of “nowhere dense” to show space of real sequences with only a finite # of nonzero terms is NOT complete?

Suppose that I define my space $V$ to the the space of real sequences with only a finite number of nonzero terms. Then, I define $V_n = (a_1,a_2,\ldots,a_n,0,0,\ldots)$. Then, it is that $V$ has a ...
3
votes
0answers
50 views

Does this characterize the operator norm of the inverse?

Let $A$ be an invertible operator (bounded with bounded inverse). Then $$\frac{1}{\|A^{-1}\|} = \inf\left\{\frac{\|Av\|}{\|v\|} : v \neq 0\right\}$$ I believe I have a proof as follows, but I just ...
5
votes
1answer
76 views

Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
2
votes
1answer
53 views

Showing that the set of functions in $C(I)$ which are monotone on some nontrivial subinterval of $I$ is of first category in $C(I)$.

Let $I = [0,1]$ and let $C(I)$ be the metric space of continuous functions on $I$ with the $L^{\infty}$ norm. I am trying to show that the set of functions in $C(I)$ which are monotone on some ...
4
votes
1answer
97 views

Condition for degenerate eigenvalues for a matrix

Given a diagonalizable matrix $M$ (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ...
0
votes
3answers
30 views

Some questions of vectors and dense subsets

I have a couple of quick functional analysis related questions: 1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. ...
1
vote
2answers
32 views

A question about the quotient isomorphism

Let $X$, $Y$ be two vector spaces and $f: X \rightarrow Y$ be a surjective map. If $M\subset $ker$f\subset X$ is a subspace, and $X/M$ is isomorphic to $Y$ (it is induce by $f$), can we conclude that ...
1
vote
3answers
39 views

$C^0([a,b])$ is an infinite dimensional vector space

I am proving that $C^0([a,b])$ is an infinite dimensional vector space. The fact that it is a vector space is clear. But I cannot understand how to prove that it has infinite dimension. Let ...
1
vote
1answer
27 views

Inverse of an $L^\infty$ matrix

Suppose that $A(\underline{x})\in\mathbb{R}^{n\times n}$ is a positive-definite matrix whose entries $A_{ij}\in L^\infty(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open domain, $n>1$. Is ...
1
vote
1answer
30 views

is this equality true?

Let $\{M_\alpha\}$ be a family of closed subspaces of a Hilbert space H. Is this equality true? ${(\cap M_\alpha^\bot)^\bot}=\overline{span(\cup M_\alpha)}$. thanks for your help.
3
votes
1answer
33 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
2
votes
3answers
47 views

Prove that if $\dim X'<\infty$ then $\dim X<\infty$

I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$. How can I prove this? I ...
1
vote
1answer
63 views

Calculate C*-subalgebra generated by $A$, $A^*$ and $\mathbb 1$

I've given a matrix $A=\left( \begin{array}{ccc} 1-3 \cos (2 \lambda ) & 3 i \sin (2 \lambda ) & 2 i \sin (\lambda ) \\ -3 i \sin (2 \lambda ) & 3 \cos (2 \lambda )+1 & 2 \cos ...
0
votes
2answers
30 views

Bessel-(LIKE) Inequality

Let $H$ be the Hilbert space, and let $M_1,M_2,...,M_n$ be mutually orthogonal closed linear subspaces of $H$. If $P_{M_i}x=x_i$, then show that $$\sum\limits_{i=1}^n\|x_i\|^2\leq\|x\|^2 ,$$ The ...
2
votes
1answer
80 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
1
vote
1answer
48 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...