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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89 views

Show that the semigroup S(t) here described is a contraction semigroup

Definition 1: Let $H$ be a Hilber space. A semigroup is a family $\{S(t)\}_{t \ge 0}$ of continuous linear operators $S(t): H \rightarrow H$ such that (i) $S(0)=I$, where $I$ is the identity operator; ...
0
votes
1answer
29 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
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1answer
12 views

About the weighted Sobolev norm

I'm wondering that the Sobolev norm with weight $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ is equivalent to the norm $$ ...
6
votes
1answer
450 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
5
votes
4answers
223 views

is $T$ compact operator?

is $T$ compact operator? $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ where $t\in[0,1]$ with supremum norm Could you please help.
2
votes
1answer
44 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
1
vote
0answers
77 views

Ascoli-Arzela Theorem [duplicate]

Wikipedia gives the Ascoli-Arzela theorem for a Compact Hausdorff Space. Following through the proof sketch at the end of the linked section, I could not find where the Hausdorff property is used. The ...
4
votes
2answers
50 views

Reflexivity of $\ell^p$

I'm having bad difficulties in understanding how to prove that $\ell^p$ with $1<p<\infty$ are reflexive spaces. Every text I have consulted give that as a trivial result because "observing that ...
1
vote
1answer
57 views

If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$. I ...
1
vote
1answer
32 views

If $\lVert u(t) \rVert \leq \lVert u_0 \rVert$ for solution of PDE, is $\lVert u(t) \rVert \leq \lVert u(s) \rVert$ for all $t \leq s$?

Suppose I have some nonlinear PDE on some time interval $[0,T]$ $$u_t + A(u) = 0$$ $$u(0)=u_0$$ and I have managed to show existence and uniqueness of solution with $\lVert u(t) \rVert \leq \lVert u_0 ...
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 ...
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
2
votes
2answers
68 views

Stone's theorem

I have some basic doubts about Stone's theorem. 1) Can we apply Stone's theorem to conclude that given any Unitary operator U, we can find a self adjoint operator A such that U = exp(i A). That is, ...
0
votes
1answer
31 views

NOT bounded operator

$C^{1}([0,1]; \mathbb{R}),C([0,1];\mathbb{R}) $ endowed with the norm $\|u\|=\sup_{x \in [0,1]}|u(x)|$ \begin{eqnarray} Dom(T_{a})&=&C^{1}([0,1];\mathbb{R}),\, ...
0
votes
0answers
19 views

Weak star convergent sequence and $L^\infty(0,T;L^\infty(\Omega))$

Suppose $u_n$ is uniformly bounded in $L^\infty(0,T;L^\infty(\Omega))$ where $\Omega$ is a bounded domain. From this answer, we know that the dual space of $(L^1(0,T;L^1))^* = ...
3
votes
2answers
67 views

Dual of $l^\infty$ is not $l^1$

I know that the dual space of $l^\infty$ is not $l^1$, but I didn't understand the reason. Could you give me a example of an $x \in l^1$ such that if $y \in l^\infty$, then $ f_x(y) = ...
2
votes
1answer
28 views

Weak-star lower semicontinuity in $L^\infty$

Let $u_n \rightharpoonup^* u$ in $L^\infty(\Omega)$. Do we get something like $$\lVert u \rVert_{L^\infty} \leq \liminf_{n \to \infty} \lVert u_n \rVert_{L^\infty}$$ i.e. a weak-star lower ...
2
votes
1answer
64 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
-2
votes
1answer
70 views

eigenvalues to Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
0
votes
0answers
24 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
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vote
0answers
49 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
0
votes
0answers
72 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
0
votes
0answers
23 views

A problem of convergence in $L^2(0,T;L^6(\Omega))$

Why $u_m\to u\mbox{ in }L^2(0,T;L^2(\Omega))\Rightarrow u_m 1_{\{|u_m|\leq\lambda\}}\to u 1_{\{|u|\leq\lambda\}}\mbox{ in }L^2(0,T;L^6(\Omega))?$
1
vote
0answers
25 views

Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$ -f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0 $$ This is what I have so far: We can show ...
3
votes
2answers
57 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
3
votes
1answer
74 views

Solve the equation: $x(t)-3\int_0^1(s+t)x(s)ds=y(t)$

Given $y\in L^2[0,1]$, Solve the equation: $$x(t)-3\int_0^1(s+t)x(s)ds=y(t)$$ I have noticed that the equation is $(I-K)(x(t))=y(t)$, where $K(f(t))=\int_0^13(s+t)f(s)ds$ is a compact integral ...
2
votes
0answers
45 views

Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
2
votes
0answers
53 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
1
vote
2answers
38 views

A question about reflexive spaces

Quoting wikipedia "a normed vector space is reflexive if it coincides with its bidual". Another definition, more precise is that a normed vector space is reflexive if its evaluation map ...
4
votes
2answers
41 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
2
votes
1answer
46 views

The dual space of locally integrable function space

I'm strongly interested in dual spaces. I learned the dual spaces of $L^p$, $L^{\infty}$, $C(X)$ and $M(X)$. I wonder the dual space of locally integrable function space in $\mathbb{R}^n$. The ...
1
vote
0answers
16 views

Eigenvalues of rotation invariant operators on 2-sphere

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore ...
0
votes
1answer
19 views

Continuous functional calculus, C*-subalgebra

Let $A$ be a unital C*-Algebra and $U\subseteq A$ a C*-subalgebra (not necessarily unital). If I take $T \in U$, is there any criterion as to when $f(T) \in U$? I mean, if $f=1$, and U is not unital, ...
1
vote
0answers
18 views

The definition of Hardy space

The definition of the Hardy space consists functions $u(z)$ that are analytic a. outside the closed unit disc or b. inside the open unit disc What is the difference between the two definition ...
2
votes
3answers
99 views

Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ \mathbb{R}$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ ...
1
vote
0answers
46 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
2
votes
1answer
34 views

A question on linearity of inner product

The linearity of inner product on $(X,\langle.,.\rangle)$ is usually written as $$\langle x+\alpha y,z\rangle = \langle x,z\rangle + \alpha\langle y,z\rangle,\qquad \forall (\alpha,x,y,z)\in R\times ...
2
votes
0answers
27 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
5
votes
1answer
70 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
1
vote
1answer
33 views

Given Tf(x), find the equivalent operator m(k)f^(k) in the Fourier transform sense.

Let $f\in L^2(\mathbb R)$, let $$ g(x) = Tf(x) = \int^{x+1}_{x} f(s)ds $$ Find $m\in L^\infty(\mathbb R)$ such that $\hat g(k)=m(k) \hat f(k)$. Use this to show that $T$ is a bounded linear operator ...
0
votes
0answers
76 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
2
votes
1answer
23 views

Another proof of the iniectivity of a linear operator

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed ...
0
votes
1answer
42 views

What is the smallest non-trivial Hilbert space?

I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof? One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
1
vote
3answers
47 views

What kind of function space does the a set of linearly independent exponential form?

I'm confused because you can define a basis $\{1, x, x^2,\ldots\}$ as the basis for the space of polynomials, $\{1, \sin(x), \cos(x),\ldots\}$ as the basis for fourier series, but little words are ...
2
votes
0answers
38 views

Convergence of Step Function Defined by Averages

For a function $f \in L^2[0,T]$, and a uniform partition $P = \{0=t_0, t_1, \ldots, t_n = T\}$ of the domain, we can define a step function approximation as the average value over each interval in the ...
1
vote
1answer
28 views

Bounded Functions: Uniform Closure

Let $E$ be a Banach space. Consider the space of bounded functions $\mathcal{B}:=\{f:\Omega\to E:f\text{ bounded}\}$ equipped with the supremum norm ...
2
votes
1answer
58 views

Show solvability of ODE without explicitly calculating solution

Show that $$ u + u^{(4)} - u^{(2)} = f $$ has a solution $u \in H^4(\mathbb R)$ (without explicitly calculuting it) for every $f \in L^2(\mathbb R)$! What criteria for solvability for such ODE's ...
8
votes
2answers
405 views

Banach space valued integration (Riemann type)

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
1
vote
1answer
18 views

weak convergence of probability measures and unbounded functions with bounded expectation

Assume that $\mu^n$ are probability measures on $R$ that convergence weakly(-*) to $\mu$, i.e for all $f \in C_b (R)$ (bounded and continuous), we have that $\int f(x) \mu^n(dx) \rightarrow \int f(x) ...
1
vote
1answer
36 views

what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set ...