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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).

-1
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1answer
17 views

Compact linear operator on normed space

I want solution of this question.... Show that $T\colon l^2\to l^2$ defined by $Tx=y=(\eta_j)$, $\eta_j=\xi_j/2^j$ is compact. (https://i.stack.imgur.com/lxvHS.png)
0
votes
0answers
13 views

Proof of an equality

Show that $(1+r)(-(1+r)^{-1}-(1+r)^{-2}-...-(1+r)^{-T+1})'_r = \frac {1+r}{r^2} (1 - \frac{Tr+1}{(1+r)^{T}} )$
0
votes
1answer
13 views

Prove that an integral operator is compact

Let $H=L^2([0,1])$ (Lesbegue-integrable in L^2) and $T:H \rightarrow H$ is defined as: $Tu(t) = \int_{0}^t 4s^3u(s)ds$, t in [0,1] I have showed that T is linear and continuous but have no idea how ...
3
votes
2answers
123 views

Geometric Hahn Banach implies Analytic Hahn Banach.

I want to prove that the geometric Hahn Banach theorem implies the analytic one. Edit: To avoid confusion I will state the vesion of H.B theorems im familiar with: Analytic H.B: Let $X$ be a ...
0
votes
0answers
9 views

In a complex metrizable topological vector space, $d(0,\alpha x)\neq |\alpha|d(0,x), \ \alpha \in \mathbb C.$

Let $(X,\tau)$ be a complex metrizable topological vector space with the metric $d$. Does the following hold: $$d(0,\alpha x)=d(0,x),\ \forall \alpha \in \mathbb C, |\alpha|=1 \ ?$$ In general, the ...
0
votes
0answers
14 views

Biorthogonality of trigonometric functions

I have to make such ${ g }_{ n }(f)$ (where is $f\in { L }^{ P) }(0;\pi )$) that $${ g }_{ n }(\cos { mx } )={ \delta }_{ nm }$$ In another words , ${ \left\{ cos(nx) \right\} }_{ n\in { N }_{ 0 } ...
2
votes
0answers
28 views

Prove under two hypothesis that $\left\|\sum_{k=1}^dA_k^*A_k \right\|=\left\|\sum_{k=1}^dA_kA_k^* \right\|$

Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$. For an arbitrary $(A_1,...,A_d) \in \mathcal{L}(E)^d$, the equality $$\...
0
votes
0answers
12 views

Multiplication of cyclic diagonals.

We denote by $D_k$ the set of all matrices of the form $A=(a_{i,j})_{i,j=0}^{n-1}$ , such that $a_{i,j}\neq 0$ for $i-j=k$ or $k-n$, and $k=1,2,\cdots n-1$. Then we have to show that if $A\in D_k$ and ...
0
votes
0answers
20 views

Strong continuity of $\langle Au,v \rangle=\int u^3 v dx$

I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by $$\langle Au,v \rangle=\int u^3 v dx$$ is strongly (weak to ...
4
votes
1answer
105 views

basis of neighborhoods of zero in Schwartz Space

I have the following question: In $S(\mathbb{R}^n)$, and for $\alpha, \beta\in \mathbb{N}^n $, we defined \begin{align} ρ_{\alpha,\beta}(\phi) = \sup_{x\in\mathbb{R}^n}|x^{\alpha}D^{\beta}\phi(x)...
1
vote
1answer
26 views

Why does $\int_{-1}^1 ((1-x^2)(P_m'P_n-P_n'P_m))'\,dx = 0$ for Legendre polynomials?

I was looking at a proof of the orthogonality of the Legendre polynomials in Lebedev's Special Functions and their Applications: I can't understand why the integral of the first term vanishes. I ...
7
votes
1answer
46 views

If $E$ is Banach and $E^*$ is its dual, is every $T:E^*\rightarrow E^*$ an adjoint?

If $E$ is a (complex) Banach and $E^*$ is it's dual, is every bounded $T:E^*\rightarrow E^*$ an adjoint? I am mostly interested in when $T$ is an automorphism. If not, is $\{T: E^*\rightarrow E^* \ |...
0
votes
2answers
27 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 ...
1
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0answers
37 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 ...
2
votes
1answer
33 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
2answers
44 views

What is the definition of a bounded operator in an infinite dimensional Hilbert Space?

I am struggling to understand the meaning of a bounded operator in a Hilbert Space. Does a bounded operator simply means that if it acts on an element of the Hilbert Space, the "result" is bounded?
4
votes
1answer
35 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 ...
0
votes
0answers
16 views

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

I have to prove that, given $f\in C^{1, \frac{\alpha}{2}}([a, b])$, $f\geq0$ and $\alpha\in(0, 1)$, there exists a constant $K$ such that $$ \|e^f-1\|_{1, \frac{\alpha}{2}}\leq K\|f\|_{1, \frac{\...
0
votes
1answer
33 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 ...
2
votes
1answer
37 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
30 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 ...
3
votes
1answer
47 views

Is a Rational Rotation Algebra a Cutdown of a Matrix Algebra?

Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2\pi i \...
0
votes
1answer
20 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}^{\...
1
vote
1answer
53 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^...
0
votes
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 ...
3
votes
0answers
24 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) :...
0
votes
1answer
28 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 $[0, \infty)$. ...
1
vote
2answers
29 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 ...
1
vote
0answers
27 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} ...
2
votes
1answer
117 views

weak* topology is not defined by any translation invariant metric when $X$ is infinite dimensional

There is an exercise in Folland's real analysis, page 170 If $X$ is an infinite dimensional Banach space, then the weak* topology is not defined by any translation-invariant metric. He gives a ...
2
votes
3answers
109 views

If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? (I'...
2
votes
3answers
60 views

Proof that the following map $\Phi:\ell^1\to(\ell^\infty)'$ is not surjective

I am working on the dual spaces of sequence spaces, and I want to show that the map $$ \Phi:\ell^1\to(\ell^\infty)',\qquad(\Phi y)(x)=\sum_{i\in\mathbb{N}}y_ix_i $$ is not surjective. I have already ...
1
vote
0answers
46 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 ...
1
vote
2answers
31 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 $f$)? ...
3
votes
0answers
25 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 ...
6
votes
1answer
60 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 ...
0
votes
2answers
47 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, $...
8
votes
2answers
10k views

Differentiability implies Lipschitz continuity

Let $f:[0,1]\to\mathbb{R}$ be a continuous function and suppose $f$ is differentiable at $x_0\in [0,1]$. Is it true that there exists $L>0$ such that $\lvert f(x)-f(x_0)\lvert\leq L\lvert x-x_0\...
1
vote
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}^...
1
vote
0answers
28 views

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 ...
2
votes
2answers
27 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 $\psi=(\psi_n)_{n\geq 1}\in \ell^\infty$ ...
2
votes
1answer
34 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 ...
3
votes
1answer
29 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 ...
1
vote
0answers
29 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,) ...
0
votes
1answer
20 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: $$\...
3
votes
2answers
76 views

Show that a complex function differentiable in all $\mathbb C$ is constant if it is bounded.

Given $X$ a complex normed space, and $f: \mathbb{C} \rightarrow X$ a function that is differentiable at all the points of $\mathbb{C}$. Show that if $f$ is bounded, then $f$ is constant. Any idea ...
6
votes
1answer
148 views

Hölder continuity definition through distributions.

I am trying to prove that for a given Hölder parameter $\alpha \in (0, 1)$ and a distribution $f \in \mathcal{D}'(\mathbb{R}^d)$ the following are equivalent: $f \in C^{\alpha}$ For any $x$ there ...
2
votes
1answer
23 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{...
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^\...