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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1answer
51 views

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
0
votes
1answer
34 views

For two positive operators on Hilbert space is it true that $A \ge B \implies \|A\| \ge \|B\|$?

$H$ is Hilbert space. $A$ and $B$ is positive linear operators from $H$ to $H$ i.e. $\forall x\in H\, (Ax,x),\,(Bx,x)\ge 0$. $A\ge B$ means that $A-B$ is positive. Does that means that $\|A\| \ge \|B\|...
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. ...
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 ...
1
vote
1answer
79 views

approximating an $ L^1 $ function with a function of compact support.

Can we approximate an $L^1$ function of several variables $ f(x_1, x_2,.., x_N) $ with a continuos function $ g(x_1, x_2,.., x_N) $ of compact support in sense of $ L^1 $ $\quad $ ? That is for $\...
1
vote
0answers
10 views

Convergence of operators and evaluation map for functions with values in a locally convex space

Let $E$ be a locally convex Hausdorff topological vector space, and $U$ a domain in $\mathbb{R}^n$. Suppose that we are given a continuous linear map $T: E \to C^{\infty}(U) \otimes E$. (The space $C^{...
2
votes
1answer
39 views

Norm of convolution

Let $f,g: \mathbb{R}^n \to R$. Let $|| \cdot ||_T$ be a translation invariant norm on functions on $\mathbb{R}^n$. How can I prove that $||f*g||_T \leq ||f||_1 ||g||_T$ (where $f*g$ means convolution ...
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 ...
9
votes
2answers
193 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
2
votes
0answers
50 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 ...
3
votes
1answer
67 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
0
votes
0answers
25 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 ...
2
votes
0answers
60 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
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 ...
1
vote
0answers
71 views

Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
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
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 ...
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?
6
votes
1answer
120 views

The set of w*-continuous operators is closed for the weak* topology?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
4
votes
1answer
31 views

$A$ has a countable dense subset. How to describe a possible countable dense subset in $M_n(A)$?

Let $A$ be a C$^*$-algebra and $M$ be a countable dense subset in $A$. Let $M_n(A)$ be the $C^*$-algebra of $n\times n$-matrices with entries in $A$, $n\in\mathbb{N}$. Then $M_n(A)$ should have a ...
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
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 ...
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}\...
2
votes
1answer
49 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 ...
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\;\;\;\...
5
votes
1answer
70 views

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

As the question title suggests, why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite? Here, we do not assume that the measure is ...
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 $T:=\iota\iota^\ast$ ...
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$\...
0
votes
1answer
51 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
1
vote
0answers
17 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 $f\land 1\in F$ (...
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
32 views

Sufficient (and necessary?) conditions for normality of Gaussian process integral

Question Let $X(\cdot)$ be a Gaussian process on $\mathcal{J}=[a,b]\subseteq\bar{\mathbb{R}}$ (extended real line) with mean $\theta(\cdot)$ and covariance $\Sigma(\cdot,\cdot)$. My goal is to find ...
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 \...
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 ...
2
votes
2answers
65 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 ...
0
votes
0answers
34 views

Integral involving Dirac delta composition

I am trying to evaluate the integral $$\int dx \ \ f(g(x)) \ \delta(\alpha-g(x))$$ where $\alpha$ is a constant and $g$ some invertible function. Here's what I did: a change of variables $$g(x)=y$$...
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 ...
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 ...
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
1answer
98 views

Infimum of lower semicontinuous functions

The following proposition is from the book Nicolae Dinculeanu Integration on Locally Compact Spaces: Let $H$ and $K$ be two compact Hausdorff spaces and $\alpha$ a continuous mapping of $H$ onto $K$. ...
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 $T\in B(X,Y)$
5
votes
3answers
144 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
33 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
114 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 $f(0)$...
-8
votes
0answers
156 views

Show that map is norm preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...