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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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2answers
28 views

does the definition of continuity require that the domain is the reals?

When we are talking about continuity at $c$. We say for a given epsilon, there is a distance delta such that for all $x$ within this distance of $c$, $|f(x)-f(c)|<\epsilon$. What if there are some ...
2
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1answer
34 views

Difference of Closed Convex sets in Banach Space

Let $A$ be a closed, convex, set in a Banach space $X$, and let $B$ be a closed, bounded, convex set in $X$. Assume that $A \cap B = \emptyset$. Set $C = A- B$. Prove that $C$ is closed, and convex. ...
1
vote
1answer
36 views

Proving an inequality between $(1, \frac{\alpha}{2})$-Hölder norms of two functions

I have to prove that, given $f\in C^{1, \frac{\alpha}{2}}([a, b])$, such that $\|f\|_{\infty}<L$ for some $L$, $f\geq0$ and $\alpha\in(0, 1)$, there exists a constant $K$ such that $$ \|e^f-1\|_{1,...
0
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1answer
34 views

Is the norm $p(x) = \max_{t \in [0,2]}|x(t)| + \left(\int_0^1|x(t)|^7\right)^{1/7}$ on $C[0,2]$ induced by any scalar product?

I have a norm on space $X = C[0,2]$: $$p(x) = \max_{t \in [0,2]}|x(t)| + \left(\int_0^1|x(t)|^7\right)^{1/7}$$ Is that norm induced by any scalar product? I try to find counterexample for ...
0
votes
1answer
32 views

Uniqueness of solution for $a(u,v)=F(v)$

Let $a(u,v)$ be a bilinearform on a hilbert space $\mathcal{H}$ which satisfies all conditions for the Lax-Milgram Lemma. Furthermore, $$a(u,v)=F(v),\ \forall v\in\mathcal{H}$$ for a bounded ...
0
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1answer
21 views

understanding part of proof in Banach-Steinhaus theorem

Theorem: If a sequence of linear bounded operators $\{A_n\}_{n=1}^{\infty}$ is a Cauchy sequence in every point of the Banach space $E_x$, then the sequence of norms $\{\lVert A_n \rVert\}_{n=1}^{\...
4
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1answer
42 views

Compact embedding of the domain and compact inverse

I have several problems in showing this point of a problem: we consider $X$ Banach space and $T: D(T) \to X$ a closed operator with domain $D(T) \subseteq X$. Let be $T$ bounded, invertible and ...
3
votes
0answers
28 views

Adjoint of a polynomial in a closed linear operator.

Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T: $ P(T) :...
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1answer
39 views

$T$ is closed $\iff$ for arbitrary $\{ x_n \}\in D(T)$ such that $x_n\to x,$ and $Tx_n\to y,$ we have $x\in D(T)$ and $Tx=y.$

Let $X$ and $Y$ be normed linear spaces and $T:X\to Y$ be any map. Then, $T$ is closed if and only if for arbitrary $\{ x_n \}\in D(T)$, domain of $T$, with $x_n\to x,$ and $Tx_n\to y,$ we have $x\in ...
0
votes
1answer
29 views

Banach-Steinhaus (Uniform-boundedness theorem) application

Let $X$ and $Y$ be Banach spaces. Consider a family of linear bounded operators $\{L_{\alpha}\}_{\alpha \in J} \subset \mathcal{B}(X,Y)$ where $J \neq \emptyset$ is a given subset of $[0, \infty)$. ...
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vote
0answers
28 views

Computing the derivative of an inner product

I want to differentiate (1) wrt the vector $x \in \mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix \begin{align} \frac{1}{2} g(x)^T w(x) g(x) &&&& (1) \end{align} ...
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0answers
43 views

If $f$ is $\beta$-smooth and non-negative, then $|f'(x)|^2\le 2\beta f(x)$?

I am reading a paper, and I found this conclusion from a proof. I am wondering why we can conclude that if a function $f$ is $\beta$-smooth and non-negative, then $|f'(x)|^2\le 2\beta f(x)$. A ...
1
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0answers
47 views

Uniform Boundedness Principle for Functionals

Kindly check if my proof is correct. Alternative proofs are welcome too! Let $\Delta$ be an arbitrary index set and let $E$ be a complete metric space and $\{ f_\alpha \}_{\alpha \in \Delta}$ be a ...
3
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0answers
35 views

Fredholm operators on non-Banach spaces.

Apparently Fredholm operators are usually (at least in Wikipedia and my functional analysis lecture) only defined as operators $T$ between to Banach spaces $X$ and $Y$. As far as I can see, the ...
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0answers
13 views

Reference for compact embedding for Holder space on $\mathbb{R}^n$

Suppose $0<\alpha<\beta$, and $\Omega$ is a bounded subset of $\mathbb{R}^n$. Then the Holder space $C^{\beta}(\Omega)$ is compactly embedded into $C^{\alpha}(\Omega)$. But if $\Omega=\mathbb{R}^...
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0answers
40 views
+50

Help proving orthogonal compliment of nullspace of adjoint operator is the closure of the range

I'm supposed to show that $N(A^{*})^{\perp}=\overline{A(X)}$. I've already shown that $N(A^{*})=A(X)^{\perp}$, and I've used that in my proof. I feel like my proof is not complete. Some of the ...
1
vote
1answer
57 views

Prove that space is Hilbert

Let $$H_0^1(0,1)=\{f\in W^{1,2}(0,1):f(0)=0\}$$ and a norm $$\| f\|=\left (\int_0^1 |f'(x)|^2\mathrm{d}x\right )^{1/2}$$ be given. I want to show that if a sequence $(u_n)_{n\in\mathbb{N}}$ in $(H_0^...
6
votes
1answer
61 views

K time differentiable function

Is there any k time differentiable function such that $$f(f'(f''(f'''(......f^{(k)}(x))))=x$$ for all $x$ belongs to $\mathbb R$? EDIT:- What will the case be when the order of the functions taken ...
2
votes
2answers
30 views

Bounded sequence perpendicular to dense subset of $\ell^2$

Consider the real Banach space $\ell^2$ of square summable sequences and let $\mathcal{A}\subset \ell^2$ be a dense subspace. Suppose I have a bounded sequence $\psi=(\psi_n)_{n\geq 1}\in \ell^\infty$ ...
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0answers
33 views

self-adjoint bounded generates analytic semigroup

Engel Nagel A Short Course on Operator Semigroups Corollary II.4.8 states: (There should be a typo. If $\delta=0$ then the spectum is empty, but normal operator has a non-empty spectrum? Anyways,) ...
2
votes
1answer
21 views

Continuous distribution-valued function induces distribution

Suppose that the map $\mathbb{R}^n \to \mathcal{D}'(\mathbb{R}^n), \hspace{3mm}\eta\mapsto E_\eta$ is continuous. Furthermore let $\mu$ be a Radon-measure with compact support. I'm having trouble ...
0
votes
1answer
21 views

Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$

Let $(s_n)_{n\in\mathbb{N}} \subseteq \mathbb{R}$ such that for all $n$: $0 < s_n \leq \frac{1}{n} $. Let $p>1$. How to show that the space of sequences $ l^p_s := \{x \in \mathbb{C}^\mathbb{...
1
vote
1answer
20 views

Are these distributions the same?

Consider the following distribution, where $\delta$ is the Dirac delta: $$f(x,y)=\delta(x)+\delta(y).\tag1$$ This can be viewed as a limit of the following sequence of smooth functions: $$\...
0
votes
2answers
50 views

If $X$ is a real normed linear space, then $B(0,r)=r B(0,1)$

If $X$ is a real normed linear space, I want to prove that $$B_{r}(0)=r B_{1}(0)$$ My proof Let $z\in rB(0,1)$, there exists $x\in B(0,1)$ such that $z=rx$. $$\|z-0\|=\|rx-0\|=r\|x-0\|<r$$ So, $...
2
votes
1answer
45 views

Discuss strong and weak convergence of a sequence in $W^{1,p}$ Sobolev space

Discuss the strong and weak convergence of the sequence of functions $$u_n(x)=\frac{1}{n}\sin nx+2\sqrt{x}$$ in the $W^{1,p}(0,1)$ Sobolev space. Pointwise limit is $u(x)=2\sqrt{x}$ and can ...
0
votes
1answer
23 views

Construction of a function from $\mathbb R^{\mathbb R^2}\to \mathbb R^\Gamma $ where $\Gamma =\mathbb R^2/_\sim$.

Let $\sim$ an equivalence relation, and denote $\Gamma=\mathbb R^2/_\sim$ the quotient space. We denote $\pi:\mathbb R^2\to \Gamma $ the natural projection. And now I want to identify $\mathbb R^\...
2
votes
1answer
24 views

Superposition of of bump functions identically equal to 1.

I am trying to create a superposition of bump functions that adds identically to 1. Specifically I am looking to add two bump functions, say $f(x)$, $h(x)$ and $g(x)$ so that if $I,J,L \subset \mathbb{...
2
votes
1answer
27 views

Does the von Neumann algebra generated by a normal operator contain all commuting projections?

Let $H$ be a Hilbert space and $T\in B(H)$ a bounded normal operator. Let $\mathscr{A}$ be the von Neumann algebra generated by $T$. Is it true that $\mathscr{A}$ contains every orthogonal projection ...
3
votes
1answer
34 views

Continuity of Energy Functional

Let $u : \Omega \times [0,T]$ be a function such that $u \in C^{2,1}(\Omega \times [0,T])\cap C^{1}((0,T);L^{2}(\Omega))\cap C([0,T);H_{0}^{1}(\Omega))$ for $\Omega \subset \mathbb{R}$ an unbounded ...
8
votes
1answer
37 views

“bounding” an unbounded operator

I was wondering if, given a certain unbounded operator on a Hilbert space, it can (naively speaking) be "cutted" (or "bounded") by certain projections. So, thinking about this in a more sensible way, ...
1
vote
2answers
35 views

Converges in inner product space

Assume $(f_i)_{i\in I}$ is an orthonormal/orthogonal system in an (complex) inner product space. Does $$\sum_{i\in I}\langle f_i,f\rangle f_i$$ always converges for any $f$ (may not to $f$)? ...
2
votes
1answer
35 views

Sectorial operator: $0\in \rho(A)$ or $0\not\in \rho(A)$?

I am confused with a characterisation of the infinitesimal generators that generates analytic semigroups. In the following characterisation do we or do we not need the origin (or if the sector is ...
0
votes
1answer
23 views

Understanding the defintion of dual operators

I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators. I'm having some difficulties understanding the following definition - Why $A^*$ is $Y^*\...
2
votes
0answers
40 views

Subgradient of Entropy

If $(X,\Sigma\,u)$ is a finite measurable space, define the map $$ \begin{aligned} X& \rightarrow (-\infty,\infty)\\ T(f)&\triangleq \int_{x \in X} f \log(f)\nu(dx), \end{aligned} $$ where $X\...
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0answers
21 views

A “multi-range” version of Banach-Steinhaus Theorem for Fréchet Spaces

A general form of the infamous Banach-Steinhaus theorem for Banach spaces can be stated as follows: Let $X$ be a Banach space, $\{Y_\alpha \}_{\alpha \in \Lambda}$ Be a collection of normed spaces. $...
2
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0answers
25 views

Duality pairing in Banach spaces

Suppose we have a Banach space $X$ and its dual $X^{'}$. Then the duality pairing is often written $$\langle f,v\rangle_{X^{'},X} = f(v)$$ for $f \in X^{'}$ and $v \in X$. But is it allowed to ...
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votes
0answers
26 views

Projection from point onto plane

Let the plane is $\prod:\vec{x}\bullet\hat{n}=d$ be a plane, where $\vec{x}\in R^{3}$ and $d\in R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the ...
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0answers
22 views

find extrema min,max on a multivariable function

I have to evaluate min,max inside and on the boundary of a domain: $D=\{xy-1\le 0,|y-x|\le1\}$ $f(x,y)=(y-x)e^{xy}$ So That's a ($xy-1$) hyperbola and two lines. I proceeded like so: for $y=1+x $ ...
0
votes
2answers
40 views

Let $\alpha >1.$ Then $\forall x\gt 0: \psi(\alpha x)\leq \alpha \psi( x)\;.$ True or False?

Let $ \psi$ be a function satisfying : $\psi: \mathbb{R}^+\rightarrow \mathbb{R}^+$ . $\psi $ is non-decreasing. $\psi (x)< x, \forall x> 0$. I want to know if the following statement is ...
0
votes
1answer
41 views

If $X$ is a real normed linear space and $r>0$, then $B_{r}(x_0+y_0)=B_{r}(y_0)+\{x_0\}$ for fixed $x_0,\,y_0\in X.$

Let $X$ be a real normed linear space and for $r>0$, let $$B_{r}(x_0)=\{x\in X:\|x-x_0\|\leq r\}.$$ I want to prove that $B_{r}(x_0+y_0)=B_{r}(y_0)+\{x_0\}$ for fixed $x_0,\,y_0\in X.$ My attempt ...
1
vote
2answers
31 views

How is the dual cone of a subspace its orthogonal complement?

From Boyd and Vandenberghe's Convex Optimization: A dual cone of a subspace $V \subseteq \Bbb R^n$ is it's orthogonal complement. $V^{*} = \{y : v^Ty = 0, \forall v \in V\}$ but the dual ...
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1answer
18 views

Compact open operator between Banach spaces

Let $X,Y$ be Banach space, $Y$ infinite dimensional. Show that no $T \in \mathcal{K}(X,Y)$ is open. By definition $T$ is open if and only if $\exists r >0$ such that $B_Y(0,r) \subset T(B_X(0,1))$ ...
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vote
2answers
28 views

Prove or Disprove: Summation of two functions (at least one discontinuous) supports IVP, if both of them support IVP.

Let, f and g be two functions on R, support IVP and at least one them is discontinuous. Then prove or disprove ( with example) whether f+g also supports IVP. If f and g, both are continuous, then it ...
0
votes
1answer
57 views

Prove that if $F: X \longrightarrow C(Y, Z)$ is continuous, then $f : X \times Y \longrightarrow Z$ is continuous. [on hold]

The space $C(Y, Z)$ is supposed to be a topological space with topology of uniform convergence. I wanted to prove the fact using the definition of continuity, but I've failed. I really don't ...
0
votes
3answers
37 views

Rudin's functional analysis appendix A4 (a)

Quick question about the following theorem: If $K$ is a closed subset of a complete metric space $X$ then the following three properties are equivalent: (a) $K$ is compact (b) Every ...
0
votes
3answers
43 views

Compact operator on $L_2([0,1],m)$

Consider the Hilbert space $H=L_2([0,1],m)$ where $m$ is the Lebesgue measure on the interval $[0,1]$. Let $T \in \mathcal{L}(H,H)$ given by \begin{equation*} T\ f(x)=x \ f(x) \ \ \ \ f \in H,\ x \...
1
vote
1answer
33 views

Compactness in weak$^*$-topology on $l_1(\mathbb{N})$

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). Is $\mathrm{co}(\mathrm{Ext}(K))$ (the convex hull of its extreme points) compact in the ...
1
vote
1answer
73 views

How to find Nash equilibrium in pure and mixed strategies? Problem.

A two player game. Players choose a number from a segment $[0;1]$. A payoff function of the first player $f_1(x,y)=|x-y|,$ where $x$ and $y$ - the numbers chosen by the first and the second players ...
7
votes
1answer
57 views

If $\mu$ has a density with respect to the Lebesgue measure, is $C_c(\mathbb R)$ dense in $L^p(\mu)$?

Let $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$. Is $C_c^\infty(\mathbb R)$ dense in $L^p(\mu)$ for all $p\ge1$? Let $\lambda$ denote the Lebesgue measure on $(\mathbb R,\...
3
votes
2answers
58 views

Compact operator $L:\ell^2\to\ell^2$ with $\Vert L\Vert=1$ such that $\Vert L(x)\Vert<\Vert x\Vert$ for all $x$

Let $\ell^2$ denote the space of square summable sequences of complex numbers. Let $L:\ell^2\to\ell^2$ be a linear operator with $\Vert L\Vert=1$ such that for all $x\in\ell^2\setminus\{0\}$, $\Vert L(...