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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5 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
2
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0answers
13 views

Is $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$ dense in $H^1(\Omega)$?

Can it be true that the space $$\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$$ is dense in $H^1(\Omega)$? If so, please give me a reference to this. Every $u \in H^1$ has $\Delta u \in ...
2
votes
1answer
30 views

When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
3
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1answer
11 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
4
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5answers
151 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
-1
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0answers
18 views

a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ and Kolmogorov-Riesz compactness theorem

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{d}$ , $\mathcal{F}$ a set of all probability densities $f$ such that $\mathcal{F}$ is a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ ...
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1answer
38 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
1
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0answers
17 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
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0answers
21 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
2
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2answers
28 views

extend a linear function

Let $P$ denote the subspace of $C^0([0,1])$ defined by polynomials restricted to [0,1]. Suppose that $l:P\rightarrow \mathbb{R}$ is a linear function with the property that $p(x)\geq 0$ in $x\in ...
1
vote
1answer
41 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
0
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0answers
29 views

Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
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0answers
34 views

Intersection of the closure of epigraph and subgraph of f function

Can you give an example of a function $f:R^p \to\ R$ such that $Grf \neq \overline{epif}\cap\overline{subf}$ where $p\in N$ and $Grf=\{(x,y)\in R^px R \mid f(x)=y \}, epif=\{(x,y)\in R^pxR \mid ...
2
votes
1answer
31 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
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0answers
26 views

Boundedness of linear operator and weak convergence

Let us assume that $X,Y$ are Banach spaces and $T : X \to Y$ is a linear operator. Show that: $T$ is bounded $\Leftrightarrow$ for any sequence $ \{ x_n \}^{\infty}_{n=1} $ weakly convergent to some ...
2
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0answers
47 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
1
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0answers
33 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
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0answers
35 views

Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at ...
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0answers
9 views

weak* convergence of periodically extended function

Let $U := \Pi^d_{i=1}(a_i,b_i) \subset \mathbb{R}^d \ (a_i<b_i \ \text{for each} \ i )$ and let $f \in L^p(U)$ for some $1<p<+\infty$. Let us extend $f$ periodically on whole $\mathbb{R}^d$ ...
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0answers
23 views

weak* convergence for sequence in $ L^\infty$

Let $ \Omega \subset \mathbb{R}^d $ be a bounded and open set. Suppose $ \{f_n\} \subset L^{\infty} (\Omega) , f \in L^{\infty} (\Omega) $. Prove that $ f_n \rightharpoonup^* f \ \ \text{in} \ \ ...
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0answers
33 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
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0answers
53 views

What is the “actual definition” of the following?

Imagine you were standing on the ground and as the ground starts moving you stay exactly where you are. Your change in movement, by standing on a moving surface, is like a path-line. Suppose we take ...
13
votes
3answers
276 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
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1answer
29 views

how to find norm of the operator Ax(t) = cos (t)*x(t)?

Please explain me how can I get norm of this operator: $Ax(t)=\cos(t)x(t)$ where $A\colon C[-\pi/2,\pi/2] \to C[-\pi/2,\pi/2]$. Thanks
6
votes
3answers
37 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
1
vote
1answer
35 views

Trace map from $H^1$ into $H^{\frac 12}$, does this statement imply another?

Consider trace map $T:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ on a sufficiently smooth domain $\Omega$. It has a partial inverse $E$. If we have the statement $$F(u,Eu) = 0\quad\text{for all ...
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1answer
87 views

Determinant: Continuity

Reference Build-up on: Determinant: Definition Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its determinant $\det:\mathcal{L}(V)\to\mathbb{C}$. Introduce a norm ...
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0answers
26 views

C*-algebras: States?

I'd like to better understand states on C*-algebras. What properties should I investigate and in which order? (Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what ...
3
votes
1answer
41 views

Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.

Edit: Prove that if $u,v \in H^{1}(\mathbb{R})$ then $uv \in H^{1}(\mathbb{R})$. My idea is to approximate with functions in $C^{\infty}(\mathbb{R})$ with compact support. Let $u,v \in ...
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0answers
29 views

$c_o$ is not isometric to $c_0 \oplus c_0$

$c_0$ is the Banach space of sequences converging to zero and $c_0 \oplus c_0$ is its algebraical direct sum with itself equipped with the norm $||(\xi,\eta)|| := ||\xi||+||\eta||$. How to prove that ...
2
votes
1answer
31 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
3
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4answers
176 views

Determinant: Alternative Definition (Matrices)

Reference Foundation for: Determinant: Continuity Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. The rank of an endomorphism: ...
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0answers
17 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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0answers
22 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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0answers
24 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
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2answers
26 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
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0answers
22 views

Problem 2.7-9 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
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1answer
9 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
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2answers
33 views

Knowing a function while only knowing its partial derivatives?

So again we study a physics course without studying mathematics course We are in the work energy chapter , and I'd like to know if you can know the function $f(x,y,z)$ if you know all of its partial ...
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1answer
31 views

$C_c(\mathbb R^n)$ is not dense in $\mathcal L^\infty(\mathbb R ^n)$

I'm having some difficulties in manipulating the space $\mathcal L^\infty(\mathbb R ^n)$, and I want to show that $C_c(\mathbb R^n)$ is not dense in $\mathcal L^\infty(\mathbb R ^n)$, but I can't find ...
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0answers
23 views

Exercise on functional analysis

I have a problem with this exercise of functional analysis about weak convergence: Let $U:=\prod_{i}^{d}(a_{i},b_{i})$ a subset of $\mathbb{R}^d$ with $a_{i}<b_{i}$ for each i and let ...
2
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0answers
23 views

Separable Hahn-Banach and the axiom of choice

We had in our functional analysis course a proof for the Hahn-Banach theorem on a separable Banach space which doesn't need, according to our professsor, the axiom of choice. Yesterday I read the ...
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0answers
23 views
+50

Gateaux variation (Functional Derivative) of functional with convolution

Given the following functional $F[f]=\int f(x) \log(g(x)) dx$ find Gateaux variation. Also, $g(x)$ is given by convolution $g(x)=y(x) * f(x)=\int y(\tau) f(x-\tau) d\tau$, so \begin{align*} ...
2
votes
1answer
40 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
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0answers
15 views

Geometric intuition behind of uniformly rotund in every direction

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $\lim_{n\to\infty} ||x_n-y_n||=0$ whenever $x_n, y_n \in S_X$ are such that $\lim_{n\to\infty} ||x_n+y_n||=2$ and ...
0
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0answers
18 views

The resolvent of a differentation operator on $C[a,b]$

Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$. I have to find $R_A(\lambda)$ for ...
-1
votes
1answer
47 views

Fundamental theorem of calculus with Gâteaux differentials and Riemann integrals

Let $f:[a,b]\to E$ where $E$ is a Banach space and let $Df(x,h)$ be its Gâteaux differential in $x$ with direction $h$. If $\mathbb{R}\to E$, $h\mapsto Df(x,h)$ is linear and continuous, then we write ...
3
votes
1answer
60 views

Having trouble understanding a proof after it applies the Hanh Banach theorem.

I have been reading a proof on the convergence of Newton's method that has been fairly easy to follow except for a single step that has totally mystified me because it suddenly depends on a lot more ...
0
votes
0answers
13 views

Multivariable calculus of variations (second variation)

So I understand that, if the multivariable Euler-Lagrange equations are satisfied, then the first variation is zero. But that doesn't mean we really are dealing with a minimum of the action or a ...
1
vote
1answer
23 views

Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that ...