Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

Let $m$ be a positive integer and define $$ Lf = -\frac{d}{dx}(1-x^{2})\frac{df}{dx}+\frac{m^{2}}{1-x^{2}}f $$ on the domain $\mathcal{D}(L)\subset L^{2}(-1,1)$ consisting of all twice ...
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1answer
41 views

Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$

Let $X$ be a normed linear space and $M$ be a proper closed linear subspace of $X$. Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$ My Work: Let $ ϵ>0$. Since ...
2
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2answers
46 views

prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$

Let $X$ be a linear normed space over $\mathbb{C}$. If a linear functional $L$ on $X$ is not continuous, prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$ Clearly $\{Lx:\|x\|\leq 1 \}\subseteq \mathbb{C}$....
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1answer
34 views

Expected value is affine in distribution

How to show that $E_{P_X}[X]$ is affine in $P_X$. That is for two distributions $P_{X,1}$ and $P_{X,2}$ we have that \begin{align*} E_{\alpha P_{X,1}+(1-\alpha) P_{X,2}}[X]=\alpha E_{ P_{X,1}}[X]+(1-\...
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1answer
43 views

Minkowski functional $p_E$ is continuous if and only if $0\in E^0=\text{int}E$

Let $X$ be a normed topological vector space. Prove $p_E$ is continuous $\iff 0\in E^0$. In the above $p_E(x)=\inf\{t\ge0: x\in tE\},$ with $E$ an absorbing set $E\subset X$ is the Minkowski ...
2
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1answer
57 views

Let $(X,d)$ be a metric space and $f:X\to X$ a function, is $d(x,f(x))$ a lower semicontinous function?

So I was trying to prove that if $f$ satisfies a special property the the function $d(x,f(x))$ is lower semicontinous but then I couldnt come up with a counter example of the following statement: Let $...
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1answer
42 views

Proof that v belongs to l_p space under certain conditions

I am struggling with the following problem: Let $M >1$ and $\lambda \in (0,1)$, $\mathbf{z} \in \mathcal{l}_p$. If $|v_t|^p < (M\sum_{s=t}^\infty \lambda^{s-t+1} |z_s| )^p $, then $\mathbf{v} \...
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2answers
45 views

If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? (I'...
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1answer
76 views

The existence of a translation-invariant Borel measure on a set of Lipschitz continuous functions

Fix a positive constant $K$, and let $T$ be the set of functions from $[0,1]$ to $[0,1]$ that are Lipschitz continuous with constant $K$ or less. $T$ is a closed convex subset of $\mathcal{C}([0,1])$ -...
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1answer
83 views

Which of the followings have a fixed point?

Consider the following sets : $$S=\left\{(x,y)\in \mathbb R^2:x^2+y^2=1\right\}.$$ $$D=\left\{(x,y)\in \mathbb R^2:x^2+y^2\le 1\right\}.$$ $$E=\left\{(x,y)\in \mathbb R^2:2x^2+3y^2\le 1\right\}.$$ ...
2
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1answer
115 views

Dual Spaces and Topological Vector Spaces

I have a question regarding dual spaces. Before, let me write that this all issue looks really problematic to me, and I already touched it quickly in another question. However, in that occasion, the ...
2
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2answers
125 views

Orthonormal Sets and Compactness

1) Let $\{u_n\}$ $(n=1,2,\ldots)$ be an orthonormal set in Hilbert space $H$. Show that this set is closed and bounded but not compact. 2) Let $Q$ be the set of all $x\in H$ of the form $$x=\sum\...
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votes
1answer
27 views

Compactness of translation operator in weighted spaces

Let $x,v\in\Bbb R^d$, $t\in \Bbb R$ and $m(x,v)$ be a smooth strictly positive function rapidly decaying on infinity - think $m(x,v) = \exp(-|x|^2-|v|^2)$. Define Banach spaces $X$ and $Y$ by $$\|h(x,...
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1answer
71 views

Linear functionals which share the properties of the integral

In my multi-dimensional real analysis class, we have recently defined the definite integral over $\mathbb{R}$ as so: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be continuous. Define the support ...
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1answer
53 views

Why if $T$ is not a bounded operator then exists $ (x_n) $ that converges to $ 0_{X} $ for which $ \| T(x_n) \| \geq n^2 $ for all $ n $?

Let $X$ and $Y$ be normed spaces. Suppose that $ T: X \to Y $ is a linear operator and assume that $T$ is not bounded. Why with these assumptions can I say that exists a sequence $ (x_{n})_{n \in \...
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1answer
89 views

Showing $f=0$ almost everywhere

Let $\psi_n(x)=e^{-x^2/2}P_n(x)$ where $P_n$ is a degree $n$ polynomial with real coefficients. Assume that $$\int_{\mathbb{R}}e^{-x^2/2}P_n=0.$$ Suppose that for any $f\in L^2$, such that $$\int_\...
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1answer
34 views

Trouble in a proof , functional analysis

So, here is the point in the proof that I don't understand, he uses that it holds for f and g which are boundered(limited, e.g. there is some M such that $|f|<M$ and $|g|<M$) then for all p, $1\...
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1answer
39 views

Nash inequality : does $f\in L^1$ and $\nabla f \in L^2$ implies $f\in L^2$?

Let $f$ be any function that belongs to $L^1(\textbf{R}^d)\cap H^1(\textbf{R}^d)$ ($d$ a positive integer). Nash inequality applies in this case and gives us $$\| f\|_{L^2}\leq C \| f\|_{L^1}^r \| \...
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1answer
90 views

A consequence of the open mapping theorem

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have ...
2
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1answer
63 views

Norm of a linear map, rational numbers, continuous functions

Let $\{ x_n \}$ be an injective sequence of all rational numbers in $[0,1]$. Let $\mathcal{C} ([0,1])$ with complex values be equipped with the norm: $$||f|| = \sqrt{\sum_{n=0}^{\infty} 2^{-n} |f(x_n)...
2
votes
1answer
210 views

Resolvent: Norm

Given a Banach space. Consider a closed operator: $$T:\mathcal{D}(T)\to E:\quad T=\overline{T}$$ Due to the Neumann series it holds: $$R(\lambda):=(\lambda- T)^{-1}:\quad\|R(\lambda)\|\geq\frac{1}{d(...
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1answer
66 views

Strictly convex iff norm is strictly sub additive

Show that the closed unit ball in a normed linear space is strictly convex i ff the norm is strictly sub additive. One part is easy strictly sub additive implies strictly convex, but I'm not able ...
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1answer
97 views

Dual space of weighted $L^p(\omega)$

Let $\omega \in A_p$, where $A_p$ is the family of Muckenhoupt weights. I'm wondering what is the topological dual space of $L^p(\omega)$. Is it isometrically isomorphic to $L^q(\omega)$? (1/p + 1/q = ...
2
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1answer
34 views

Are ideals generated by separable subspaces separable?

Suppose that $X$ is a compact Hausdorff space and take a sequence $(f_n)$ in $C(X)$ such that the ideal generated by $(f_n)$ is proper. Must this ideal be separable as a Banach space? It looks to me ...
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1answer
63 views

Approximating the gradient of a function by $L^2$ functions (cut-offs)

Let $u:(0,\infty) \to \mathbb{R}$ be such that $u' \in L^2(0,\infty)$, but $u \notin L^2(0,\infty)$. However $u \in L^2(0,T)$ for all finite $T$. Is it possible to find a sequence of functions $u_n \...
2
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2answers
44 views

$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy \leq ||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $

We are given the following, $$ b:\mathbb R^d \times \mathbb R^d \rightarrow \mathbb R,\;\; f:\mathbb R^d\rightarrow \mathbb R $$ and $$ f\in L^2(\mathbb R^d)\; ,\;b\in L^2(\mathbb R^d\times \mathbb R^...
2
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1answer
46 views

How do I show that this function is increasing, and has discontinuities at points in this sequence

I have a series $\sum c_n$ that converges, and $c_n>0$. Then $(x_n)$ is a sequence in $(0,1)$. Let $f(x)=\sum\limits_{n=1}^\infty c_nf_n(x)$ where $$f_n(x) = \begin{cases} 1 & : ...
2
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1answer
55 views

Compact operator as certain limit

Let $H$ be an infinite-dimensional Hilbert space with basis $\{e_i\}_{i=1}^\infty$. Let $P_n := \sum_{i=1}^n e_ie_i^*$, i.e. $P_n$ is the projection onto the span of the first $n$ basis vectors. Let $...
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votes
3answers
256 views

Derivative of an integral with respect to a function

Can some one help me out this derivative: $$ \frac{\partial\int_{-\infty}^{\infty}f(x)g(x)dx}{\partial g(x)} $$ Appreciate any explanation! Many thanks to those who answered or commented on my ...
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1answer
136 views

Algebraic and orthogonal complements

I have been trying to understand this Encyclopedia of mathematics article. Specifically, in the comments section there is the following comment: The codimension of a subspace $L$ of a vector space $...
2
votes
2answers
46 views

$X+Y$ is closed $\Leftrightarrow$ $\|x\|\leq c\|x+y\|$ for all $x\in X$ and all $y \in Y$.

The problem says Let $(Z,\|\cdot\|)$ be a Banach space. Let $X$ and $Y$ be two closed subspaces of $Z$ such that $X\cap Y=\{0\}$. Prove that $X+Y$ is closed if, and only if, there exists $c\geq 0$ ...
2
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1answer
88 views

How to find the norm of $ \Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$ in $\ell^2$

Suppose that $(x_n)$ is a sequence in $\ell^2$, i.e. $\displaystyle \sum_{i=1}^{\infty} x_n^2 < \infty$. Define: $$\Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$$ Find $\| \Lambda \|_2$ ...
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1answer
72 views

Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ ...
2
votes
1answer
36 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank 1....
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1answer
97 views

Counterexample Poincaré Inequality for $H_0^1$ in 2D

Is there any counterexample to the Poincaré inequality $$\int_\Omega|f|^2dx\leq C(\Omega)\int_\Omega|\nabla f|^2dx $$ for $f\in H_0^1(\Omega)$, $C(\Omega)>0$ and $\Omega\subset\mathbb{R}^2$? I ...
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60 views

How to prove the closure $\bigcup_{n\ge 1} F_{n}$ is totally bounded and closed

Let $(X,p)$ be a metric space. Write $F$ for the set of subsets of $X$ which are closed, bounded, and non-empty. For each integer $n\geq1$, write $F_{n}$ for the set of subsets of X which are finite ...
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1answer
30 views

Closure of intersection with vector subspace

I am confused with the footnote on page 198 of http://www.ma.huji.ac.il/~razk/iWeb/My_Site/Teaching_files/TVS.pdf Essentially: Let $X$ be a topological vector space and $Y$ a finite-dimensional ...
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2answers
43 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: $$F_n:\Omega\to\...
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2answers
69 views

Spectral Measures: Unitary Map [duplicate]

This thread is a record. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}\to\mathcal{H}:\quad N^*N=NN^*$$ and its spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\...
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1answer
45 views

Are $(l^1, \|.\|_2)$, $(l^2, \|.\|_3)$ Banach spaces?

Let $l^1=\{(x_n)_n|\, \sum\limits_{n=1}^{\infty}|x_n|<\infty\}$ with norm $$\|x\|_p=\left(\sum\limits_{n=1}^{\infty}|x_n|^p\right)^{1/p}.$$ Is $(l^1, \|.\|_2)$ a Banach space? Is $(l^2, \|.\|_3)$...
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1answer
113 views

Compact operator space is the greatest ideal of $B(H)$

Suppose $H$ is a separable infinite dimensional Hilbert space. Show that if $A\in B(H)$ is noncompact, then there exist two operators $B,C$ such that $BAC=1$. Clearly if $A$ is invertible it holds, ...
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1answer
101 views

Equivalent characterizations for Reflexive spaces

Well I'm reading about Reflexive spaces those days and I would like to see a proof for two different claims. The first claim is that a Banach space is reflexive iff every bounded functional attains ...
2
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1answer
51 views

States: KMS-Condition

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state $\omega$. Does it suffice to have on a dense set the KMS-condition: $$F(t+i\beta)=\lim_nF_n(t+i\beta)=\omega(\tau^t[B_n]...
2
votes
1answer
108 views

On Fredholm operator on Hilbert spaces

Let $u: H \to H'$ be a continuous linear operator and $H,H'$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite ...
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1answer
51 views

parameter operator $A_a$ is compact??

I need some help in this exercise. Let define operator on $ L^2[0,1]$: $$ A_af(x)=\int_{0}^{1}{|x-y|}^{a-1} f(y)dy $$ for f $\in L^2[0,1] $. Prove that A_a is compact for all $a>0$. I see that ...
2
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1answer
117 views

How to prove that a function is irrational?

I need to know how to prove that a given function is irrational. Examples: $$ f(x)=\sqrt{1+x^2} $$ $$ f(x)=\tan(x) $$ Information about the definition of rational and irrational functions would be ...
2
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1answer
304 views

Sum of closed subspaces of a Hilbert space is closed

Let $M, N ⊂ H$ ($H$ Hilbert), be two closed linear subspaces. Assume that $\langle u, v\rangle = 0$ $∀u ∈ M$, $∀v ∈ N$. Prove that $M + N$ is closed. Take a sequence $(g_n)\in M+N$ such that $g_n\to ...
2
votes
1answer
87 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
2
votes
2answers
66 views

Fock Space: NESS

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: $...
2
votes
1answer
50 views

How to prove $\mathcal{L}e^{-U}g = e^{-U}\mathcal{L}^*g$ for Fokker-Planck operators

Let $U$ be sufficiently smooth, $\beta$ a constant and $$ \mathcal{L}p = \frac{1}{\beta}\Delta p + \nabla\cdot(p\nabla U)\\ \mathcal{L}^*g = \frac{1}{\beta}\Delta g - \nabla g \cdot\nabla U. $$ Now ...