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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).

2
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1answer
65 views

If $T(t)$ is a semigroup on $E$ and $F$ is a subspace of $E$ such that $T(t)$ is $F$-preserving, how are the generators on $E$ and $F$ related?

Let $E$ be a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ be a semigroup on $E$ and $(\mathcal D(A),A)$ denote the generator of $(T(t))_{t\ge0}$. If $F$ is a closed subspace of $E$ and $T(t)F\...
0
votes
1answer
20 views

Proof of Lemma preceding Principle of Condensation of Singularities

Under the Wikipedia page for the Principle of Uniform Boundedness, we have the Corollaries of the Uniform Boundedness Principle. The third of these relates to the Principle of Condensation of ...
3
votes
1answer
35 views

What are the approximate eigenvalues of the right shift operator $R$ on $\ell_\infty$

I have shown that the spectrum of $R=\{z\in C||z|\leq 1\}$. Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $\forall |z|=1$, $z$ is an approx. eigenvalue. However, ...
2
votes
0answers
33 views

Best constants for Gagliardo–Nirenberg inequality in the case: p=q=2

Given a function $u$ with compact support and a bounded area $\Omega\subset\mathbb{R}^n$ with $n \geq 3$. We have the well know Gagliardo–Nirenberg inequality $$ \|u\|_{L^2} \leq C \|Du\|_{L^2}. $$ ...
4
votes
1answer
77 views

Two PDE for one unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions. My ...
26
votes
1answer
1k views

Does convergence of polynomials imply that of its coefficients?

Let $\{p_{n}\}$ be a sequence of polynomials and $f$ a continuous function on $[0,1]$ such that $\int\limits_{0}^{1}|p_{n}(x)-f(x)|dx\to 0$. Let $c_{n,k}$ be the coefficient of $x^{k}$ in $p_{n}(x)$. ...
1
vote
1answer
17 views

If $\Gamma(f):=\frac a2|f'|^2$, are there $0\le\eta_k\in C_c^\infty$ with $\eta_k\uparrow1$ and $\Gamma(\eta_k)\le1k$?

Let $$\Gamma(f):=\frac a2|f'|^2\;\;\;\text{for }f\in C_c^\infty(\mathbb R)$$ for some $a>0$. How can we show that there is a $(\eta_k)_{k\in\mathbb N}\subseteq C_c^\infty(\mathbb R)$ with $$0\le\...
1
vote
1answer
27 views

Weakly convergence

If $g \in L^p(\mathbb{R})$ be a given non-trivial function, show that following sequences converge weakly in $L^p$ but not strongly in $L^p$. (a) $g_k(x)=k^{1/p}g(kx)$. (b) $h_k(x)=g(x+k)$. I need ...
0
votes
0answers
27 views

Uniform Bound on Infinitely Differentialble functions

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Problem Show that if $f$ ...
1
vote
2answers
32 views

Does a bounded linear operator on a Hilbert space conjugate to its adjoint?

Let $H$ be a Hilbert space over $\mathbb R$ or $\mathbb C$ and $f\in B(H)$. I wonder if there is a $g\in B(H)$ such that $fg=gf^*$, where $f^*$ is the adjoint of $f$. We know a matrix over a field is ...
0
votes
0answers
25 views

Estimate approximation error of a function

It is supposed to approximate a function $f$ on the interval $[a, b]$ by a function $p$ that fits piecewise a polynom of degree $n$. I've noted the following steps: Decompose $[a, b]$ into $N$ ...
5
votes
2answers
108 views

$\Delta u=3u$ then $u\equiv0$

I have the following question in which it is easy to use Fourier transform to get the answer if the function is nice enough, for example $u\in C_{0}^{\infty}(\mathbb{R}^{n})$, however here $u$ is only ...
3
votes
1answer
28 views

A subspace of a dual Banach space $X^*$ is norming if and only if its weak$^*$ closure contains a multiple of the unit ball of $X^*$

Definition A subspace $Z$ of a dual Banach space $X^*$ is said to be norming if there exists $c>0$ such that $$\sup_{f\in B_Z}|f(x)|\geq c||x||$$ for every $x\in X$ (where $B_Z$ is $B_{X^*}\cap ...
3
votes
2answers
26 views

$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$ if $H$ is Hilbert and $P,Q$ orthogonal projections.

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ orthogonal projections. Show that : $$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$$ Attempt-Thoughts : $(\Rightarrow)$ Let $PQ = 0$. ...
1
vote
0answers
41 views

Computing Euler Lagrange Equation for a Certain Functional

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where ...
-1
votes
2answers
29 views

Sequence in $l_1$

Let $x_n$ be sequence in $l_1$. $$\sum \frac{|x_i|^2}{2^i} \leq {(sup|x_i|})^2$$ True or false? I can not show this.
1
vote
1answer
37 views

What's going on with this inequality?

Let $(\mathsf{X}, \mathcal{X})$ be a measurable space, $(\mathcal{F}_b(\mathsf{X}, \mathcal{X}), \Vert \cdot \Vert_{\infty})$ be the space of all bounded measurable functions on this space equipped ...
1
vote
0answers
14 views

There exists a continuous inverse of $(\text{id}-A)$ in the set $(\text{id}-A)(X)$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$ (means that $A$ is a compact operator). Suppose that $(\text{id}-A)$ is $"1-1"$. Show that the operator $(\text{id}-A)$ has a ...
1
vote
0answers
30 views

If a closable linear operator satisfies the nonnegative maximum principle, does the same apply to its closure?

Let $E$ be a locally compact Hausdorff space and $(\mathcal D(A),A)$ be a closable linear operator on $C_0(E)$. If $(\mathcal D(A),A)$ satisfies the nonnegative maximum principle$^1$, does the same ...
1
vote
0answers
34 views

Construction of a Partition

Can someome help me to proe the following result: There exist two functions $\chi $ and $\phi$ valued in the interval $[0, 1],$ belonging respectively to $\mathcal{C}^{\infty}_0((0, \frac{1}{2}))$ ...
2
votes
1answer
23 views

Show that $\exists u_0 \in C : g(u_0) = u_0$, if $g$ is nonexpansive over a Banach subspace.

Exercise : Let $X$ be a Banach space, $C \subseteq X$ compact and convex and $g : C \to C$ a nonexpansive operator. Show that $\exists u_0 \in C : g(u_0) = u_0$. Thoughts : In a previous exercise, ...
0
votes
1answer
20 views

Heat semigroup representation

It is known that the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega),$ (with $\Omega$ being a open subset of $R^n$) generates a $C_0-$semigroup in $L^2(\Omega)$). Moreover, in ...
2
votes
0answers
67 views

Lemma 2.4.1 Introductory Functional Analysis - Kreyszig :Confirmation of my understanding

I see that questions relating to this lemma have been asked here and here . But my question is quite different in the sense , that i believe that i have understood what the author intends to say but ...
1
vote
1answer
29 views

What is the closure of the set of continuous bounded real valued functions on $\mathbb{R}^d$ under pointwise convergence?

What is the closure of the set of continuous bounded real-valued functions on $\mathbb{R}^d$ under pointwise convergence? How might one go about finding this closure? Also, are there common names for ...
-1
votes
0answers
13 views

Weak compacity and reflexive subset

Is that true that any closed subspace of a weakly compact set is also weakly compact ? Can we have that : A closed subspace of a weakly compact is reflexive ?
2
votes
0answers
16 views

If $R_\lambda$ is the resolvent of a linear operator $A$ at a regular value $\lambda$, what is $R_\lambda(\lambda-A)$?

Let $E$ be a $\mathbb R$-Banach space, $(A,\mathcal D(A))$ be a linear operator and $\lambda\in\mathbb R$ such that $$A_\lambda:=\lambda\operatorname{id}_{\mathcal D(A)}-A$$ is injective, $A_\lambda\...
4
votes
0answers
43 views

Showing that $\inf \{\|u-g(u)\| : u \in C \} = 0.$

Exercise : Let $X$ be a Banach space and $C \subseteq X$ be closed, convex and bounded. Moreover, let $g:C \to C$ be a non-expansive operator, meaning that : $$\|g(u) - g(v) \| \leq \|u-v\| \; \...
3
votes
1answer
38 views

If $f\in C_0(E)$ with $\inf_{x\in E}f(x)<\infty$, then there is a $x\in E$ with $f(x)\le\min(f,0)$

Let $E$ be a locally compact Hausdorff space and $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\}.$$ Let $f\in C_0(E)$ with $$\inf_{x\...
2
votes
1answer
21 views

$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ are orthogonal projections, then show that : $$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$$ Seeking a formal ...
0
votes
3answers
47 views

Trouble reading this proof of characterization of continuous mappings by open sets.

Here's the first part of the proof—the part I have a question about. Simply put, I don't see anything here precluding $x_0$ from being on the boundary of $S_0$. And if that's the case then $N_0 \...
1
vote
0answers
18 views

Linear functionals and complex polynomials

Question: Let $K\subset \mathbb{C}^N$ be compact and let $\mathcal{P}(K)$ denote the closure of $\{p|_K:\, p \in \mathbb{C}[z_1,\dots,z_N]\}$ with respect to $||\cdot||_K$. Show that for each non ...
1
vote
2answers
28 views

Question regarding Weierstrass theorem generalized to Stone-Weierstrass

Weierstrass theorem. Lef $f$ be a defined and continuous function in $[a,b]$. Given $\epsilon>0,$ there exists a polynomial $P$ such that $\vert f(x)-P(x)\vert<\epsilon,$ for all $x\in[a,b].$ ...
1
vote
1answer
24 views

How do I apply Banach-Steinhaus to a sequence of real numbers $(x_k)$?

How do I apply Banach-Steinhaus to a sequence of real numbers $(x_k)$? My problem is the requirement that the collection be from $E \rightarrow F$, where $E$ is Banach and $F$ is normed. Since real ...
1
vote
0answers
26 views

Generation theorem for Feller semigroups

Let $E$ be a locally compact Hausdorff space. I want to show that a linear operator $(\mathcal D(A),A)$ on $C_0(E)$$^1$ is closable and the closure $(\mathcal D(\overline A),\overline A)$ is the ...
0
votes
1answer
45 views

Conditional convergence of series $\sum_{n=1}^{\infty}a_n(\frac{k}{n}-k^{n+1}\frac{\zeta(n+1)}{n+1}),0<k\le 1$

Does the series $\displaystyle \sum_{n=1}^{\infty} a_n\left(\frac{k}{n}-k^{n+1}\frac{\zeta(n+1)}{n+1}\right),0<k\le 1$ converges conditionally given that $\sum_{n=1}^{\infty}|a_n|<\infty$. ...
1
vote
1answer
28 views

$a=cc^*c$ for some $c$. $a \in A$ a $C^*$ algebra.

Let $A$ be a $C^*$ algebra. Let $a \in A$, then there exists $c \in A$ such that $a=cc^*c$. This fact is used from example (1) of Prop 4.25. How does one show this?
1
vote
2answers
20 views

Want to show that sequences in little l 1 can be represented by orthonormal basis

I want to show that for a sequence $a=\{a_i\}^{\infty}_1 \in l^1$ I can write $$a= \sum ^\infty _{i=1} a_i e_i $$ For $\{e_i\}^\infty_1 \in l^1$ where the ith term is 1 and all else zero. I know for ...
1
vote
0answers
9 views

Functional differentiation with Grassmann variables

I want to calculate: $$\frac{\partial}{\partial (x^\rho+i\eta\, \psi^\rho)}[f_\mu(x)+i\eta\, \psi^\nu\partial_\nu f_\mu(x)]$$ where $x^\mu(\tau)$, $\psi^\mu(\tau)$ are commuting and anti-commuting ...
2
votes
1answer
20 views

Hessian of inner product

Let $f:\mathbb R^n \rightarrow \mathbb R^n$ be a $C^2$ functions and consider the function $h:\mathbb R^n \rightarrow \mathbb R$ given by $$h(x):=\langle f(x),f(x) \rangle.$$ I am wondering whether ...
1
vote
0answers
35 views

If $\mathcal D(\overline A)$ is dense, are we able to conclude that $\mathcal D(A)$ is dense?

Let $(\mathcal D(A),A)$ be a closable linear operator on a $\mathbb R$-Banach space $E$ and $(\mathcal D(\overline A),\overline A)$ denote its closure. If $\mathcal D(\overline A)$ is dense, are we ...
1
vote
1answer
48 views

Definition of closed

In topology we say that a set is $A$ is closed if its complement $A^\complement$ is open. Now if we have a set $A$ and an operation $*$ we say that $A$ is closed under this operation if $x,y \in A$ ...
1
vote
1answer
41 views

$H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$?

In a paper I see that the authors used $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary. I think that this imbedding holds ...
2
votes
1answer
23 views

How are the two versions of the Lumer-Phillips theorem given in the books of Engel/Nagel and Pazy related?

Let $E$ be a $\mathbb R$-Banach space and $(\mathcal D(A),A)$ be a densely-defined dissipative linear operator on $E$. In the book of Engel and Nagel, I've found the following verison of the Lumer-...
3
votes
2answers
80 views

Continuity of Linear Operator Between Hilbert Spaces

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Let $\mathcal{H}$ be a ...
0
votes
0answers
17 views

Set $X$ such that $2^c \leq |X|\leq |(\ell^\infty)^*|$ [duplicate]

I want to prove that $2^c\leq|(\ell^\infty)^*|$. My question is: Is there a set $X$ with $2^c\leq|X|$ such that $f:X\to(\ell^\infty)^*$ is an injective function or $g:(\ell^\infty)^*\to X$ is a ...
4
votes
1answer
57 views

Suppose $U_1,\dots,U_k$ and $V_1,\dots,V_k$ are $n\times n$ unitary matrices. Show that $\|U_1\cdots U_k-V_1\cdots V_k\|\leq\sum_{i=1}^k\|U_i-V_i\|$

Let $V,W$ be complex inner product spaces. Suppose $T: V \to W$ is a linear map, then we define $$\|T\|:=\sup\{\|Tv\|_{W}:\|v\|_{V}=1\}$$ where $\|v\_{V}\|:=\sqrt{\langle v,v\rangle}$ and $\|Tv\|_{W}...
0
votes
1answer
38 views

Show that there is a solution of the Laplace equation $(\mu-A)p=f$

Let $C_0(\mathbb R)$ denote the space of continuous functions vanishing at infinity equipped with the supremum norm, $B$ be a contractive linear operator on $C_0(\mathbb R)$ and $$Af:=\lambda(Bf-f)\;\;...
4
votes
1answer
49 views

Finite rank operators on Hilbert spaces

Let $H$ be a Hilbert space. Question 1: Are all rank one operators from $H$ to $H$ is of the form $$T:H\rightarrow H, x \mapsto \langle x,u\rangle v $$ For some $u,v \in H$. Question 2:...
0
votes
1answer
34 views

Rudin's functional analysis, theorem 4.23 (existence of a certain sequence)

If $X$ is a Banach space, $T \in \mathcal{B}(X)$, $T$ is compact, and $\lambda \neq 0$, then $T - \lambda I$ has closed range. Proof with questions below Proof: By (d) of Theorem 4.18, $\text{dim }...
1
vote
2answers
17 views

Does positive part preserve Holder continuity?

Let $u^+$ denote the positive part of the function $u$ on a bounded domain $\Omega.$ If $u \in C^{0,\alpha}(\bar \Omega)$, is also $u^+ \in C^{0,\alpha}(\bar \Omega)$?