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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2
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0answers
55 views

Frechet derivative of square root on positive elements in some $C^*$-algebra

Let $A$ - is some unital $C^*$ algebra, and $P$ is set of all strictly positive elements in $A$. We can define map $\sqrt{?} : P \to A$ which takes positive element and returns its (unique) strictly ...
1
vote
0answers
45 views

Differential $\sqrt{1+B(x,x)}$ map in $C^*$-algebra

Let $A$ is $C^*$-algebra and $P \subset A$ is subset of all elements $a \in A$ such that $a > 0$ (nonnegative) and $||a|| < \frac{1}{\sqrt{1-q}}$ (norm bounded) for some $0 < q < 1$. Let $...
2
votes
1answer
19 views

Show that $q(T)(x)=\sum_{n=1}^\infty q(\lambda_n) \langle x,e_n\rangle e_n$ coincide with $q(T)=\sum_{k=0}^n a_kT^k$

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
0
votes
1answer
75 views

How to write this as a function of only $z^2$?

$F(z)=\int_{{\mathbb R}^n}f(x)e^{2\pi x \cdot z-\pi x \cdot x-\frac{\pi}{2}z^2}dx$ Given $f\in L^2({\mathbb R}^n)$ is a radial function. $z\in{\mathbb C}^n$ and $z^2$ denotes $z \cdot z$ (dot ...
1
vote
0answers
33 views

Sufficient conditions for $f(T)$ to be compact and self adjoint whenever $T$ is compact and self adjoint

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
2
votes
1answer
50 views

Continuous function on the unit sphere [duplicate]

Let S$^2$ := $\lbrace$ x $\in$ $\mathbb{R}$$^3$ : $\Vert x\Vert$$_2$ $\rbrace$ $\subset$ ($\mathbb{R}$$^3$, $\Vert .\Vert$$_2$) and T: S$^2$ $\to$ ($\mathbb{R}$, $\vert x\vert$ ) a continuous function....
1
vote
0answers
24 views

Limit relevant to parametrised semi-group 2

Let $s\geq 1, \epsilon >0, T>0$ and $f \in \mathcal{C}([0,T], H^s(\mathbb{T}))$. Define the function $$g(t,x):= \int_0^t(Id-\exp\left(-i\tau\epsilon \Delta)\right)f(\tau,x)d\tau.$$ I want to ...
0
votes
0answers
28 views

Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function?

[Q.] Is there a semicontinuous function, which has its discontinuous set with non-zero measure? Remark: Given a semicontinuous function, the set of all discontinuous points may be uncountable, for ...
0
votes
1answer
46 views

Convergence in $L^p$ and convergence almost everywhere

Why $f_n$ converges to $f$ in $L^p$ space implies that exists subsequence of $f_n$ converging to $f$ almost everywhere?
5
votes
1answer
59 views

Exponential of Operators

Let $H$ be an Hilbert Space $\exp(T)$ the exponential for an operator $T \in L(H)$. I know that $\exp(A)^{*} \exp(A)=\exp(A) \exp(A)^{*}=id$. Can I conclude that $A^{*}A=AA^{*}$? Cannot find an ...
0
votes
2answers
30 views

Any closed convex bounded set is weakly compact in a reflexive Banach space.

Let $X$ be a reflexive Banach space. Any closed convex bounded set is weakly compact. I know it is true. But, I can't find a reference. Anyone can help?
0
votes
1answer
23 views

Neighborhood of inclusion in space of Lipschitz maps is 1-1

Let $B \subset \mathbb{R}^n$ be the closed unit ball. Let $i(x) = x$ denote the inclusion map. Let $\|\cdot\|$ be any norm. Given $f:B\to \mathbb{R}^n$, define the sup norm $\|f\|_\infty:=\sup_{x \in ...
0
votes
0answers
26 views

Textbook/monograph for microlocal analysis

I want to grasp the theory of microlocal analysis and apply this theory to some PDEs in $R^n$. But most textbooks I found put much priority on manifolds. Sadly, I know little about them and don't ...
1
vote
0answers
29 views

Limit relevant to parametrised semi-group

Let $s\geq 1$, $T>0$, $\epsilon >0$ and $f\in\mathcal{C}^1(0,T,H^{s-1}(\mathbb{T}))\cap \mathcal{C}(0,T,H^{s}(\mathbb{T}))$. Consider the propagator $\exp\left[\displaystyle-\frac{it}{\epsilon}\...
1
vote
2answers
30 views

Space of bounded linear maps induced by different norm

Suppose $X,Y$ are normed space, there are two norms $\|\|_1^Y,\|\|_2^Y$ on $Y$ which induce the same topology. We can define the norm of bounded linear mappings from $X$ to $Y$ as $$||f||_i=sup\{\|f(x)...
2
votes
1answer
37 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
2
votes
1answer
50 views

Why the space of complex measures is Banach?

I've read the proof from here: Space of Complex Measures is Banach (proof?) and understood the part that proves that constructed limit is complex measure. But the first part is a bit unclear for me. I ...
0
votes
0answers
33 views

If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
1
vote
0answers
18 views

Characterization of Banach sublattice of L^1

Let $(X, \Sigma, \mu)$ be a measure space and let $F\subset L^1(X,\Sigma,\mu)$ be a Banach sublattice of $L^1$ with the following properties: (1) If $f\in F$, $f$ real-valued, then $f\land 1\in F$ (...
3
votes
2answers
59 views

Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
0
votes
0answers
27 views

Extending a unitary operator

Suppose that $\mathcal{H}_1$ and $\mathcal{H}_2$ are two separable Hilbert spaces and that $X\subset \mathcal{H}_1$ is a dense subspace (i.e. $\overline{X}=\mathcal{H}_1$). If $\operatorname{W}:X \to \...
0
votes
1answer
27 views

Isometric embedding of $\ell^2$ into $L_1$.

Let $\{Y_n\}_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables on some probability space $(\Omega, \mathcal{F}, P)$ following a standard complex Gaussian distribution (that is, the ...
0
votes
0answers
22 views

uniqueness of weak solutions for parabolic pde Evans

Hi I am trying to understand Evan's proof on uniqueness of weak solution in chapter 7. For the proof of theorem 4 below, I can see (35) and (36) make sense. But I have difficulty to see how Gronwall's ...
1
vote
1answer
40 views

prove a bounded and linear operator

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
0
votes
1answer
89 views

Continuous injection and density in $l_p$ spaces

If $r \le s$ then $l_r$$\subseteq$ $l_s$ . How can I prove there is a continuous injection $l_r$ $\hookrightarrow$ $l_s$? The suggestion was to use the fact that $\Vert$x$\Vert$$_r$ $\le$ $\Vert$x$\...
2
votes
0answers
37 views

explicit self adjoint operator which has no diagonalization

Let a linear operator $T : H \to H$ be diagonalizable if $H$ has an orthonormal basis composed of eigenvectors of $H$ Give an example of an explicit self adjoint operator which has no diagonalization ...
2
votes
1answer
39 views

A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$

As the title says, the question is how to prove $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
2
votes
0answers
34 views

The 3rd term of the energy estimates in chapter 7 Evans PDE

Hi I wonder someone could help me check my understanding of getting an inequality of the estimate for the 3rd term $\|u'_m\|_{L^2(0,T;H^{-1}(U))}$ correctly. This inequality need to be checked is ...
5
votes
2answers
115 views

How to define $f(0)$ when $f$ is a function in $L^2$?

Any function $f$ in $L^2$ is a actually an equivalence class and has properties that only hold "almost everywhere." But it would be convenient to speak of the value of $f$ at certain points like $f(0)$...
1
vote
1answer
42 views

Comparison of capacity of sets in $\mathbb{R}^n$

This is mainly in reference to this MSE post. Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \...
1
vote
0answers
52 views

Extending a functional

Here's a question I have from an old exam without solutions. I was wondering if you could check my work (and help on part b) Suppose $g: \mathbb{R} \to \mathbb{R}$ and $\int_1^\infty |g(x)|^3 \; dx &...
3
votes
0answers
51 views

$S$ is continuous with Weak * topology from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $B(X,Y)$ [duplicate]

How to prove that prove that $S$ is weak$^*$-continuous from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $T\in B(X,Y)$ Thanks for any hints. To show that $T$ is continuous is straight forward ...
2
votes
1answer
37 views

What is the relation between the matrix of an operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \to Y$ be a linear operator. (Then $T$ is bounded since its domain is finite-dimensional). ...
0
votes
4answers
50 views

A question on linearly independent vectors in a Banach space

Given a list of linearly independent vectors $\{x_1,...,x_n\}$ in a Banach space. If for each $1 \leq i\leq n$, there is a sequence of vectors $\{y_m^{(i)}\}_{m=1}^{\infty}$ converges to $x_i$. Then ...
2
votes
1answer
57 views

$T*T$ Notation and proof

Let $T:H\to H$ be compact where $H$ is a Hilbert space and let $T^*$ be the adjoint operator of $T$. Prove that $T^*T$ is compact and self adjoint and that the eigenvalues of $T^*T$ are nonnegative. ...
0
votes
0answers
35 views

Continuous function with support continuously embedded [duplicate]

Can someone give me a solution for this? We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ ...
5
votes
3answers
145 views

Isometry map on a compact metric space

Let $X$ be a compact metric space and $f : X\rightarrow X$ such that $d (x,y)\le d (f(x),f(y))$ for all $x,y\in X$. Prove that $f$ is an isometry. I am getting stuck on this question. Can any one help ...
2
votes
0answers
19 views

Prob. 5, Sec. 4.5 in Kreyszig's functional book: The adjoint of the composite of two bounded linear operators

Let $X$, $Y$, and $Z$ be normed spaces, either all real or all complex. Let $T \colon X \to Y$ and $S \colon Y \to Z$ be bounded linear operators. Let $X^\prime$, $Y^\prime$, and $Z^\prime$ denote the ...
2
votes
2answers
66 views

A doubt about the vectorial topology on $\mathcal{D}(\Omega)$

We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the ...
2
votes
0answers
61 views

Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
1
vote
0answers
64 views

Sobolev Space dual

I'm interested in the dual space of the Sobolev space $H^1(\Omega)$ for $\Omega$ a bounded smooth domain. Of course, because $H^1(\Omega)$ being a Hilbert space, it's dual is isomorphic to itself, but ...
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votes
0answers
28 views

Showing certain map is continuous and linear.

Let $X=C[0,1]$ with norm $$\|f\|_1=\int_{0}^{1}|f(t)|dt$$ and $Y=C[0,1]$ with norm $\|f\|_{\infty}=\sup_{t\in [0,1]}|f(t)|$. Define $K: C[0,1]\times C[0,1]\to \Bbb R$ by $K(f)=\int_{0}^{1}K(s,t)...
2
votes
0answers
36 views

Fractional powers of Markov generators

Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the ...
1
vote
1answer
30 views

How to show that If $||Ax||\ge C||x||$ then $A$ is one-to-one a and range $A$ is closed? [closed]

Let $H,K$ be Banach spaces and $A$ a linear and continuous transformation from $H$ to $K$ ($A\in B(H,K)$). How do I show that if $||Ax||\ge C||x||$, then $A$ is one-to-one and range $A$ is closed? ...
1
vote
0answers
30 views

Well-definedness of Fourier transform of $f\in L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
1
vote
0answers
33 views

How to prove $n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$ is a norm on $B(H)$ and $n(T)\lt\|T\|\lt2n(T)$ where $T\in B(H)$? [closed]

Let $H$ be a Hilbert space over $\mathbb C$. If $T\in B(H)$, how to prove that $$n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$$ is a norm on $B(H)$ and $$n(T)\lt||T||\lt2n(T)\ \textrm{?}$$ I couldn'...
1
vote
0answers
56 views

energy estimates Evans PDE chapter 7

Hi I am looking at the proof of theorem 2 energy estimates in Evans PDE. I have some difficulties regarding the estimate for each term. First for the first term. Q1 I am a little vague how (23) is ...
1
vote
1answer
20 views

Applications of Positive Operator Valued Measures (POVMs)

I am wondering what some applications of POVMs are in mathematics (or mathematical physics)? I am going through Berberian's 'Notes on Spectral Theory', which shows how we can write a normal operator ...
1
vote
1answer
28 views

equivalent trace-conditions on $C^*$-algebras

Let $A$ be a $C^*$-algebra and $\tau:A\to\mathbb{C}$ linear. Claim: the following conditions are equivalent: $\tau(ab)=\tau(ba)$ for all $a,b\in A$ $\tau(x^*x)=\tau(xx^*)$ for all $x\in A$ $\tau(uau^...
0
votes
1answer
48 views

How could I search the inverse operator $(Af)^{-1}(x)$

I am try to search $A^{-1}$ when I define $A:L^2[0,2] \rightarrow L^2 [0,2] $ when $$(Af)(x)=x^{-1/4}f (\sqrt {2x}) $$ What I do: I consider that $ (Af)^{-1}((Af)(x))=Ix=x \Longleftrightarrow (Af)^{...