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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Meagre Sets: Algebra

Let meagre subsets be defined as: $A\text{ meagre}\iff A=\bigcup_{\lvert K\rvert\leq\aleph_0} A_k\text{ with }\overline{A_k}°=\varnothing$ Then it satisfies: $B\subseteq A\text{ meagre}\Rightarrow ...
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1answer
49 views

Closed range assumption in definition of Fredholm operators

There are two definitions of Fredholm operators (on a Hilbert space) that are commonly used. The first is that $\dim\ker T<\infty$ and $\dim\,\mathrm{coker} T<\infty$. An argument using the open ...
3
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2answers
92 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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1answer
97 views

Compact kernel operator on $L^p$ space

Let $\displaystyle U_1 \subset \mathbb R^{n_1}$ and $\displaystyle U_2 \subset \mathbb R^{n_2} $ measurable sets, $\displaystyle 1 < p,q < \infty $ and consider the measurable function ...
6
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1answer
51 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
2
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1answer
118 views

The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...
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1answer
26 views

Discontinuous covariance kernels

Let $I$ be a topological space, and let $c : I \times I \to \mathbb S$ denote a symmetric, nonnegative-definite function. Must $c$ be continuous with respect to the topology of $I$, or is it ...
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2answers
81 views

Prove that convex subspace of $l_2$ is compact

Prove that the convex subspace of $l_2$ consisting of all sequences $\xi$ such that $$ \sum_{n=1}^{\infty} \xi_n^2 n^2 \le 1 $$ is compact. Have no idea how to proceed, any hints or suggestions?
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1answer
367 views

Fractional Sobolev embedding into $L^\infty$

Are there any $t\in(0,1)$, $p\in[1,\infty)$ such that $W^{t,p}(\mathbb{R})$ is continuously embedded into $L^\infty(\mathbb{R})$? I have been looking several literatures, but I have not yet found ...
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0answers
21 views

Question concerning a special Fredholm operator

Let $C\subseteq H^\infty (D)$ the space of functions that are analytic in the unit disc $D:=\{z\in \mathbb{C}\ |\ |z|<1\}$, and that have continuous continuation on $D \cup$ ∂$D$ and let $C$ be ...
2
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0answers
68 views

Coercive operators are invertible

Let $V$ be a Hilbert space and let $A:V \to V^*$ be a bounded linear operator such that $$\langle Av, v \rangle \geq C|v|_V$$ for all $v \in V$. Why does this mean that $A$ is an isomorphism ...
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1answer
64 views

Proof that if $A\subset B$ then $A^* = B^*$

I prove here that if $B$ is a Banach space, and $A$ is a closed subspace of $B$, $A\subset B$, then $$A^* = B^*.$$ ($A^*$ stands for the dual of $A$.) There is obviously something wrong here but ...
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2answers
58 views

Orthogonal Projections in $U\subset V$ two subspaces in a Hilbert space

Let $U,V\subset H$ be closed subspaces of a Hilbert space $H$, and let $P_U$ and $P_V$ the respective orthogonal projections. Show: $U\subset V \Longleftrightarrow P_U=P_UP_V=P_VP_U$ Trying to ...
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1answer
52 views

Is $L^2_{\text{loc}}$ a Hilbert space?

My question is in the title. I know that $L^2$ is a Hilbert space, but I'm not sure about $L^2_{\text{loc}}$. Is it even an inner product space? Thanks.
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0answers
39 views

Decomposition of Partial Isometry

I'm reading a paper and I don't understand how the operator is being decomposed. I've tried reading about different types of decomposition but nothing I read seems relevant: (Let $\mathscr{H}$ be a ...
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1answer
29 views

Differentiability in $\mathbb R^n$

Let $U\in \mathbb{R}^n$ be open, and let $f:U\to \mathbb{R}^m$, and let $a\in U$. Let $\|\cdot\|'$ be a norm on $\mathbb{R}^n$, and let $\|\cdot\|''$ be a norm on $\mathbb{R}^m$. Prove that $f$ is ...
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0answers
35 views

The product of two projections is 0

I'm reading a paper and the paper seems to think the following is obvious: Let $S$ be a semigroup of partial isometries. Let $R = \{ E \in P(S) \cup Q(S) : E$ is minimal in $P(S) \cup Q(S)$ and for ...
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1answer
66 views

Show $T$ cannot be a compact operator

Let $(X,\lVert\cdot\Vert_x)$ and $(Y,\lVert\cdot\Vert_y)$ be normed spaces, X be infinite dimensional and $T\in\mathcal{L}(X,Y)$ Which has the property: there exists $m>0$ such that $ \Vert{T ...
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1answer
42 views

Can somebody explain the well-defined mapping to me?

Let $(X,(\langle.,.\rangle)$ be a Hilbert space over $\mathbb{K}$ with an orthonormal basis $(x_n)_{x\in\mathbb{N}}$ and let $(\lambda_n){n\in\mathbb{N}}\subset \mathbb{K}$ be a bounded sequence.The ...
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votes
4answers
211 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
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1answer
38 views

Baire: Equivalent Statements II

Moreover, why does it follow for Baire Spaces and why is it strictly weaker that: $X=\bigcup_{k=1}^\infty A_k\quad\Rightarrow\quad\exists k_0\in\mathbb{N},x_0\in X: A_{k_0}\in\mathcal{N}_{x_0}$ ...
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2answers
30 views

What happens if I define $C^0(I;X)$ like this (without caring about continuity)?

For $X$ a Banach space, let me define the space $C^0([0,T];X)$ to consist of elements $u:[0,T] \to X$ such that $$\lVert u \rVert_{C^0} := \max_{t \in [0,T]}\lVert u(t) \rVert_X < \infty.$$ So the ...
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2answers
93 views

Baire: Equivalent Statements

How to proof that these statements are equivalent: Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. ...
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1answer
64 views

How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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1answer
71 views

Is $\nabla u \in L^{\infty}$ if $u$ is bounded $C^{0}$?

I would like to prove something of the form $|A_{1}(u)| \leq c \lVert u \rVert_{L^{\infty}}$ and $|A_{2}(u)| \leq c \lVert \nabla u \rVert_{L^{\infty}}$ for some operators $A_{1},\ A_{2}$ and ...
2
votes
1answer
100 views

Find the weak sequential closure of a set in $L^2(-\pi,\pi)$

$A=\{f_{m,n}(t)|0\le m<n\}$ where $f_{m,n}(t)=e^{imt}+me^{int}$. I should find the weak sequential closure of $A\subset L^2(-\pi,\pi)$. I know what I'm supposed to do. Take a sequence in $A$ and ...
2
votes
2answers
58 views

There are $u$ in $W^{1,p}(D)$ and a subsequence $\left\{ u_{m_{k}}\right\} $ such that $\left\{ u_{m_{k}}\right\} $ weakly converges to $u$.

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $p \in (1,\infty)$, and {$u_m$} be a bounded sequence in $W^{1,p}(D)$. Then there are $u$ in $W^{1,p}(D)$ and a subsequence ...
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1answer
48 views

If $f\in L^2$, has the equation $u_{xx}=f$ an unique solution $u\in H^2\cap H_0^1$?

Let $-\infty<a<b<+\infty$ and $f\in L^2(a,b)$. Is it possible to prove that the equation $u_{xx}=f$ has an unique solution $u\in H^2(a,b)\cap H_0^1(a,b)$? If so, how can we prove it? ...
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0answers
89 views

Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ a.e. on D.

I want to prove this result: Let $D$ be an open subset of $\mathbb{R}^n$, $p \in[1,\infty)$ and $f$ be in $L^p(D)$. Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ ...
0
votes
1answer
67 views

Noncompactness of the closed unit ball in $L^2$

Let $$ L^2[0,1]=\{f:[0,1]\to\mathbb R\,\,\text{such that}\,\, \|f\|_2<∞\}, $$ where $\|f\|_2^2=\int_0^1 |f(x)|^2\,dx.$ Show that the unit sphere $$ S=\{f\in L^2[0,1]:\|f\|_2\le 1\}, $$ is ...
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votes
2answers
112 views

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...
11
votes
1answer
243 views

Inequalities on kernels of compact operators

Suppose we have a $\sigma$-finite positive measure $\mu(v)$ on $\Bbb R^d$ and we have two positive kernels on $\Bbb R^d\times \Bbb R^d$ $k_1(v,u)>0$, $k_2(v,u)>0$. We define integral operators ...
0
votes
1answer
168 views

Notation Question (Meaning of double inclusion symbols)

What does the notation $\subset \subset$ mean? In my class notes, our prof writes $\Omega \subset \subset \mathbb{R}^{n}$ to mean that "$\Omega$ is a convex subset of $\mathbb{R}^{n}$". Is that all ...
0
votes
1answer
236 views

Convergence in $L^\infty$ norm and continuous function

Let $\mathcal{C}(T)$ be the set of continuous functions on $T$, which is a metric space under the norm $\left\|f\right\|_{\infty}=\sup_{t\in T}\left|f(t)\right|$. Suppose $\{X_{n}\}$ and $X$ take ...
2
votes
3answers
128 views

Euler-Lagrange: Motivation for Definition of Weak Solutions

Let $E:M\to\mathbb{R}\cup\{+\infty\}$ be an energy functional of the form \begin{equation} E[u]=\int_\Omega L(x,u,\nabla u)dx, \end{equation} where $M$ is a subset of $W^{1,p}(\Omega)$ $(1< ...
4
votes
1answer
68 views

Do the homomorphisms really have to be continuous?

I read that If $\varphi, \psi$ are continuous homomorphisms from a normed algebra $A$ to a normed algebra $B$ then $\varphi = \psi$ if $\varphi$ and $\psi$ are equal on a set $S$ that generates the ...
11
votes
1answer
135 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
3
votes
2answers
97 views

Justifying that $S=\left\{f \in X: \int f(t)dt=0\right\}$ is compact and connected

Consider the space $X=C[0,1]$ with its usual sup-norm topology. Let $$S=\left\{f \in X: \int f(t)dt=0\right\}.$$ Justify: S is compact. S is connected.
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1answer
80 views

$\ell^p(I)$ space and a dense set of this space

Let $I$ be an infinite set and $1\leq p<\infty$. Show that $\ell^p(I)$ has a dense set of the same cardinality as $I$. For this I put $$X=\{(x_i); x_i\in \Bbb C , x_i=0 \text{ for all but ...
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1answer
75 views

Boundedness and compactness of a subset of $( \ell^2 , \|\cdot\|_2 )$

Investigate the boundedness and compactness of the following subset of $( \ell^2 , \|\cdot\|_2 )$: $$S:=\{x=(x_n) \in \ell^2 : | x_n | \le 1/\sqrt{n} \ \forall n\in N\}.$$ Can the following be ...
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vote
2answers
65 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
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1answer
44 views

Fourier series using other bases?

The theory of Fourier series, representing a reasonable function by an infinite sum of exponential functions, is very well-developed. In addition to basic functional-analytic results there are things ...
6
votes
1answer
260 views

Is Reflexivity Necessary for the Weak and Weak* Topologies to Coincide?

Let $X$ be a normed vector space, not necessarily Banach. Suppose that $X$ is not reflexive, implying the existence of such $\varphi\in X^{**}$ ($X^{**}$ being the double dual of $X$) of that for any ...
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votes
1answer
48 views

Given an arbitrary sequence {$x_n$},find a test function have the $n$-derivative equal to $x_n$ at $0$.

Given an arbitrary sequence {$x_n$}, can I find a test function have the $n$-derivative equal to $x_n$ at $0$? How to prove it?
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1answer
81 views

A basic question about function space

Im reading Carothers' Real Analysis, 1ed. Actually, Carothers has begun his talk of function space since chapter9. However, I haven't found any definition about function space. Here is a definition ...
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1answer
147 views

Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
7
votes
2answers
228 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
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vote
1answer
168 views

The Trace Class Operators Form a Banach Space

I want examining the trace class operators $L_1(H)$ of a separable Hilbert space $H$ with the norm $||A||_1=\sum\limits^{\infty}_{n=1}\lambda_n$ where $\lambda_n$ are the eigenvalues of ...
1
vote
1answer
128 views

Application of Brouwer fixed point theorem, why is compactness not required here??

Define a map $K:X_n \to X_n$ where $X=\text{span}(v_1, ..., v_n)$ where $v_i$ are basis functions some Hilbert space $H$. So $X_n$ is finite-dimensional. $B_r(0)$ denotes the ball of radius $r$ ...
2
votes
2answers
121 views

non zero linear functional and which of the following statements are true. (NBHM-$2014$)

Let $V$ be finite dimensional real vector space and let $f$ and $g$ be non zero linear functionals on $V$. Assume that $\ker(f) \subset \ker(g).$ Which of the following are true?? a. ...