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
87 views

Does scaling lead to weak convergence to the null function?

Let $f\in L^p(\mathbb{R}^d)$, with $1<p<\infty$. Is it true that $$\lambda^{\frac{d}{p}}f(\lambda x ) \rightharpoonup 0\quad \text{ weakly in }L^p\text{ as }\lambda\to+\infty?$$ One has ...
0
votes
1answer
54 views

Some questions about subspaces in Banach spaces

I just have a few question about some things in Banach spaces. Let $X$ be a separable, reflexive Banach space with basis $\{e_{i}\}$. Let $X_{n} = \text{span}\{e_{1},...,e_{n}\}$, then consider the ...
3
votes
1answer
81 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
1
vote
2answers
37 views

help me please about adjoint of operators in L1

A : L₁→L₁ 1) A x=( x₁, x₂,.....xn , 0,0,....) 2) A x= (λ₁ x₁ ,λ₂ x₂,.....) |λ n|≤1 and λ n ∈ R I need to find adjoint of operators A in given space. ...
1
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0answers
37 views

Describe all of the functions measurable w.r.t. $\mathcal{A}$ where $\mathcal{A}=\{E \subset X | E \space or \space E^c \space countable\}$

Let $(X,\mathcal{A},\mu)$ be a measure space where $X$ is uncountable, $\mathcal{A}=\{E \subset X | E \space or \space E^c \space countable\}$ and $\mu(E) = \left\{ \begin{array}{lr} ...
0
votes
1answer
47 views

operator semigroups with negative growth bound.

Let $(T(t))_{t\geq 0}$ be a strongly continuous semigroup on a Banach space $X$, for which we assume that its growth bound $\omega_0$ is negative. Let $(A,D(A))$ be the generator of $(T(t))_{t\geq ...
0
votes
1answer
42 views

help,example about disjoint operators

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$ Tx(t)=∫_0^1 tx(s)\,ds $$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what ...
2
votes
2answers
62 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
1
vote
1answer
38 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
0
votes
1answer
59 views

Am I wrong ? (2)

Let $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\infty=\sup_{s\in[0,1]}|f(s)|$$ and ...
1
vote
2answers
231 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
1
vote
0answers
31 views

Density of $C^0([0,T]\times M)$ in $L^p([0,T]\times M)$?

Let $M$ be a compact $C^2$ hypersurface embedded in $\mathbb{R}^n$ of dimension $n-1$. Is the space $C^0([0,T]\times M)$ dense in $L^p([0,T]\times M)$? How do I prove this or what theorem can I use? ...
1
vote
1answer
32 views

About definition of $L^\infty(0,T;L^\infty(\Omega))$ and null sets

The norm in $L^\infty(0,T;L^\infty(\Omega))$ is $$\text{esssup}_{t \in [0,T]}\text{esssup}_{x \in \Omega}|u(t,x)|$$ In the inner essential supremum, can the null set (on which the function fails to ...
1
vote
2answers
43 views

Proving by induction that if $\{I_k\}_{k=1}^{n}$ is a finite set of open intervals with covers $[a,b]$, then $\sum\limits_{k=1}^n |\space I_k| = b-a$?

I am trying to give an inductive proof that if $\{I_k\}_{k=1}^{n}$ is a finite set of open intervals with covers $[a,b]$, then $\sum\limits_{k=1}^n |\space I_k| = b-a$. My proof is as follows: ...
1
vote
1answer
43 views

weak form of the problem in two domains

Let $\Omega$ be an open, bounded domain, and a smooth internal boundary $\Gamma$ divides $\Omega$ into two open and connected sets, $\Omega1$ and $\Omega2$, where $\Omega1$ is strictly included in ...
0
votes
0answers
14 views

confusion about proof on Banach limits [duplicate]

I am reading a example in John N McDonald's intro to analysis and I am very confused about one line. Let $l_r^{\infty}$ denote that real linear space consisting of all bounded sequences of real ...
0
votes
1answer
121 views

Does the continuity at $0$ of the addition map in a vector space imply its continuity?

I have a question about the proof of Theorem 1.41 in Rudin, Functional Analysis, 2/e. The theorem states Let $N$ be a closed subspace of a topological vector space (t.v.s.) $X$. Let $\tau$ be the ...
0
votes
1answer
62 views

Finding resolvent kernel

For $f$ is $L^2[0,1]$ define $Kf$ by $$(Kf)(x)=\int_0^1 (1+5x^2t^2)f(t)$$ Find a function $R(x,t,\lambda)$ such that solutions to the equation $$f(x)=g(x)+\lambda \int_0^1(1+5x^2t^2)f(t)dt$$ are ...
1
vote
1answer
54 views

$(X,|.|_A)$ is Banach implies $A$ is closed

Let $(X,|.|)$ be a Banach space. We know that if $A:X\to X$ is a closed operator then $(X,|.|_A)$ is a Banach space, where $|.|_A$ is the norm defined by $$|x|_A=|x|+|Ax|$$ Then using the "continuity ...
1
vote
2answers
153 views

Constructing a sequence that is pointwise bounded but not uniformly bounded by points in a closed, nowhere dense set in $\mathbb{R}$.

I believe that this question below is asking for a sequence of functions that are bounded pointwise in $\mathbb{R}$ but NOT uniformly bounded in a closed, nowhere dense set of $\mathbb{R}$. Suppose ...
0
votes
1answer
44 views

Contraction map

I have a general question about the properties of contractive/non-contractive maps. Assuming I have a map $F$ which I know is not a contraction. Can it map a ball back to itself, i.e. for some ...
0
votes
0answers
48 views

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous?

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous? I am trying to prove continuity by limits but am failing: suppose ...
1
vote
1answer
56 views

Where am I wrong ??

Let $(X,|.|)$ be a Banach space. $A\in B(X)$ a bounded injective operator. Then we can define another norm on $X$ by $$|x|_A=|Ax|.$$ Since we have $$|x|_A\leq |A||x|$$ Then by the result of continuity ...
0
votes
1answer
63 views

Does weakly differentiable and $L^{\infty}$ imply continuity

Suppose $\Omega \subset \mathbb{R}^d$ is open, connected and bounded. Is $$W^{1,1}(\Omega)\cap L^{\infty}(\Omega) \subset C(\bar{\Omega})?$$ Here $W^{1,1}$ denotes the space of all weakly ...
5
votes
1answer
172 views

Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
-2
votes
1answer
64 views

Norm Space verification

For each of the following decide whether the suggested formula defines a norm on the indicated space. You may assume that $||f||_1=\int_0^1 |f(t)| dt$ does give a norm on the space of all continuous ...
2
votes
1answer
86 views

Is a non-compact Riemannian manifold a “measure space”?

One can define $L^p$ spaces for measure spaces with a given measure. Is a non-compact (i.e., it has a boundary) bounded Riemannian manifold a measure space? I am thinking of the manifold $(0,T) \times ...
4
votes
1answer
357 views

Pointwise a.e. convergence and weak convergence in Lp

I'm trying to prove the following theorem: Let $\{f_n\}\subset L^p(\Omega)$, $f_n \rightharpoonup f$ in $L^p(\Omega)$ ($\Omega\subset\mathbb{R}^n$ is open and bounded, $1\leq p \leq \infty$) and $f_n ...
1
vote
1answer
39 views

Convergence $ \ell^1$ sequences

if I have a sequence $(x_n) \in \ell^1$ and an element $x \in \ell^1$ and we have that for all $k \in \mathbb{N}: x_n(k) \rightarrow x(k)$, does this mean that $||x_n-x||_1 \rightarrow 0$?
2
votes
3answers
122 views

Is a function in $L^2$ which second derivative is in $L^2$ in $H^2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary. Assume that $f\in L^2(\Omega)$ and $f^{\prime\prime}\in L^2(\Omega)$. Does one have $f\in H^2(\Omega)$? Useless comments: ...
1
vote
0answers
84 views

Open ball in infinite dimensional Banach space is not weakly open

I have to prove that open ball in infinite dimensional Banach space is not weakly open. I have no idea how can I do it. I think that I should reach contradiction with infinite dimensions.
0
votes
0answers
51 views

Question about functions in Sobolev space.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. If I consider a function $g:\mathbb{R}\rightarrow\mathbb{R}$ which has the following properties: $$ |g(x)|\leq M \qquad |g(x)-g(y)|\leq ...
0
votes
1answer
26 views

If $u_n$ are uniformly bounded in $L^\infty$, $u_n \to u$ in $L^2(\Omega)$, is $u$ also bounded?

Let $\Omega$ be a bounded domain. Let $u_n \in L^\infty(\Omega)$ with $|u_n|_{L^\infty} \leq M$. If $u_n \to u$ in $L^2$, is also $u \in L^\infty$? We know that for a.a $x$, $u_n(x)$ is bounded ...
3
votes
1answer
105 views

Countable union of relatively compact sets

Let $X$ be a topological space and $\mathcal K(X)$ be $\sigma$-algebra, generated by compacts of $X$. Prove that for any set $B \in \mathcal K(X)$ either $B$ or its complement can be represented as a ...
0
votes
1answer
26 views

Proof that this operator is linear and bounded, find operator norm

A:C[-1,1] to C[0,1] That means x(0) here?
1
vote
1answer
64 views

prove that using uniform bounded theorem [duplicate]

Let $y=(\eta_j),\eta_j\in \mathbb C$, be such that $\sum \xi_j\eta_j$ converges for every $x=(\xi_j)\in c_0$ where $c_0\subset l^\infty$ is the subspace of all complex sequences converging to zero. ...
2
votes
1answer
90 views

Dual of $div$ on spaces where the tangential value is fixed

Say $\Omega$ is a domain in $\mathbb R^3$ with a smooth boundary $\Gamma$. Consider the spaces $$ H_{n,0}=\{v\in H^1(\Omega):n\cdot v \bigr |_{\Gamma} = 0\} $$ and $$ H_{t,0}=\{v\in ...
1
vote
2answers
163 views

Prove that a closed unit ball in $C[0,1]$ is not weak-compact

I have to prove that a closed unit ball in $C[0,1]$ is not weak-compact. The hint is that I should consider sets: $$V_t=\{f\in C[0,1]:|f(t)|>1/3\}$$ and $$U_t=\{f\in C[0,1]:|f(t)|<2/3\}$$ Now I ...
1
vote
1answer
47 views

Question on Gagliardo-Nirenberg.

On page 679 of this paper, the authors claim they can get a special case of Gagliardo-Nirenberg with a constant of 1/2. They prove this using functions in $C_0^\infty(\mathbb{R}^2)$, for which the ...
0
votes
1answer
95 views

Measurable and Borel Measurable Functions

Prove that if $f: X \rightarrow \mathbb{R}$ is measurable, and $g: \mathbb{R} \rightarrow \mathbb{R}^*$ is Borel measurable then $g \circ f:X \rightarrow \mathbb{R}^*$ is measurable. Could someone ...
1
vote
1answer
63 views

Compact surjective non injective operator

Let $X$ be an infinite dimensional Banach space. I know that every compact operator $A$ is not bijective or $0\in\sigma(A)$. Fox example the compact operator $A$ defined on $X=C([0,1],\mathbb{R})$ ...
2
votes
1answer
38 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
0
votes
1answer
64 views

Spectrum of an operator

Let $X=C([0,1],\mathbb{R})$ the Banach space of continuous real functions in $[0,1]$ equipped with the supremum norm. We define the operator $A$ for each $x\in X$ by $$(Ax)(t)=\int_0 ^t x(s)ds, \ \ \ ...
8
votes
4answers
708 views

is $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ compact?

Is $T$ a compact operator? $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ where $t\in[0,1]$ with supremum norm.
1
vote
0answers
21 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
6
votes
1answer
159 views

Extension of Goldstine theorem

Is the following claim true? Claim. Let $E$ be a Banach space and $F$ its closed subspace. Assume $x\in (E\setminus F)\cup\{0\}$ and $y^{**}\in F^{\perp\perp}$, then there exist a net ...
-1
votes
1answer
36 views

Some Vector Space clear ups

I'm studying for a functional analysis exit exam and I'm asked to show which of the following are vector subspaces of the vector space $F([0,1]), \mathbb{R})$ of all real valued functions on the ...
1
vote
1answer
166 views

Frechet/Gateaux differentiability of an integral operator L^2 --> R

Let $f: R \rightarrow R$ be a continuously differentiable function on the real numbers (if needed also infinitely many often differentiable). Define the Operator $F : L^2([0,1]) \rightarrow R$ for $x ...
1
vote
0answers
32 views

How to prove these equivalences?

I want to prove the following statement: Let $K$ be a compact Hausdorff space and $F\subset C(K)$. Then the following are equivalent: The closure of $F$ in the weak topology of $C(K)$ is weakly ...
0
votes
1answer
23 views

Probability space problem

Given:$( \Omega ,F,\mu)$- probability space. $f:\Omega\rightarrow X'. X$-is a Banach space, such that the mapping $\Omega\ni\omega\mapsto\langle x, f(\omega)\rangle\in L^1(\Omega,\mu)$ for all $\ ...