Tagged Questions

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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1answer
11 views

Poincare Inequality for 1-Dimensional Problem.

I am referring to the book Introduction to Functional Analysis to Boundary Value Problems and Finite Element by Daya Reddy (page ...
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0answers
12 views

A $L^1$-bounded sequence from a $H^m$-bounded sequence

I am trying to show the following: for any $m > 0$ and $\alpha \in \mathbb{N}^n$, assume $(f_j)$ is a sequence of functions which is bounded in $H^m(\mathbb{R}^n).$ Assume moreover that all the ...
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0answers
19 views

Sobolev space on closed surfaces

I was wondering if anybody here knows how the Sobolev space $H^2(\mathbb{S}^2)$ is defined? I.e. I want to integrate on this space with respect to the surface measure, but since this not the canonical ...
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3answers
82 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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1answer
40 views

Sequence in a hilbert space.

$M$ is a closed subspace of the Hilbert space $H$, and x $\in H$. Call $d = \inf_{y \in M} ||x - y||^2$ Show that there exist a sequence of elements $y_n$ of M such that $||y_n - x ||^2 \rightarrow ...
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1answer
15 views

Prove the operator on hilbert space is compact

My question is actually the same as the first part of this one, Prove that T is compact which has not been answered. I am thinking about two ways, 1) use a bounded sequence $\{g_n\}$, and try to ...
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0answers
14 views

When is $\|\phi(|T|)\|=\|\phi(T)\|$ for $T\in B(H)$?

If $T\in B(H)$, $H$ a Hilbert space, then it has a polar decomposition $T=V|T|$, where $|T|=(T^*T)^{1/2}$ and and $V$ is a partial isometry. Let $\phi:B(H)\to B(H)$ be ucp (unital completely ...
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1answer
26 views

If $T^{2}$ is a compact operator then $T$ is compact

Suppose $T$ is a bounded , self-adjoint operator on a Hilbert space such that $T^{2}$ is compact. Then prove that $T$ is compact. I proved it by continuous functional calculus but am looking for a ...
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0answers
6 views

Relation between RKHS and space of continuous functions

Consider a Mercer Kernel $K\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$, $\mathcal{X}$ being a compact subset of $\mathbb{R}^m$, and its (unique) associated Reproducing Kernel HIlbert Space ...
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1answer
25 views

The space $BPV(0,1)$ is a separable metric space under certain metric.

This is exercise 2.42 from Leoni's book A First Course in Sobolev Spaces. The BPV is defined as the space contain the function $u$ such that $$ \text{Var}[u]:=\sup\left\{ ...
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0answers
25 views

Erwine Kryszeg's _Introductory Functional Analysis With Applications_: Section 2.3, Prob. 14

Here's problem 14 in the Problem Set immediately following Section 2.3 in the book, Introductory Functional Analysis With Applications by Erwine Kryszeg. Let $Y$ be a closed subspace of a normed ...
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1answer
20 views

To bound a heat equation on a real line?

Let $\displaystyle\mathcal{H}_{t}(x)=\frac{1}{(4\pi t)^{1/2}}e^{-x^{2}/4t}$ be the Heat Kernel. The imposed initial condition for the heat equation on a real line is $u(x,0)=f(x)$ a function belongs ...
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1answer
14 views

Show that $H(\mathbb{C})$ is a Frechet space.

Let $H(\mathbb{C})=\{f: \mathbb{C}\rightarrow \mathbb{C}; f \text{ holomorphic}\}$. For each $n$ let the seminorm $p_n$ be $p_n(f)=\sup_{|z|\leq n}|f(z)|$, and let $d(f,g)=\sum_{n=1}^\infty ...
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1answer
65 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
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2answers
42 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
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1answer
30 views

Functional analysis problems collection with solutions

Could you please advise where I can find problems in functional analysis (covering more or less conventional first course in the subject) with the full solutions. The books that I have found so far ...
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1answer
12 views

Projections onto disjoint spectra in functional calculus

Theorem 6 part (i) of Lax's Functional Analysis book (Chpt 17) states (paraphrased) Suppose that the spectrum of $M$ can be decomposed as the union of $n$ pairwise disjoint closed components: ...
0
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1answer
20 views

Element with the least norm

Let $W$ be the set of points $(x_1,x_2,\ldots,x_n) \in \mathbb{C}^n$, for which $\sum_{k=1}^n x_k = 1$. Knowing that $W$ is convex and closed, find an element in $W$ which has the least norm.
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1answer
10 views

Question on Completeness of Derived Inner Product Space

Let $(\mathcal{H},\langle{,}\rangle)$ be a separable, infinite-dimensional Hilbert space. Let $\mathcal{X}''$ denote the space of bounded sequences in $\mathcal{H}$. For a Banach limit $L$, define a ...
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1answer
22 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ ...
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1answer
15 views

“Absolutely equal” linear functionals and collinearity

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb C$ and let $X^*$ denote its dual (i.e., the space of all continuous linear complex-valued functions over $X$). Suppose that $f,g\in X^*$ ...
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2answers
28 views

Continuity of vector space operations in a normed space

Here's problem 4 immediately following section 2.3 in Erwine Kryszeg's book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar ...
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1answer
49 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...
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0answers
15 views

Application of multivalued fixed point theorems in differential and integral equation. [on hold]

How does an unique solution exist in differential equation using any multivalued fixed point theorem?
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1answer
24 views

A question in functional analysis about bounded linear operator.

Suppose $Banach$ Space $E$ is the direct sum of its closed subspaces $L$、$M$, and $M$ is finite-dimensional, $T$ is a bounded linear operator from $E$ to itself. Prove that $T(E)$ is a closed subspace ...
2
votes
2answers
28 views

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space. It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the ...
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0answers
15 views

Weakly convergence w.r.t weakly closed [on hold]

If $K$ is weakly closed in a Banach space $X$,if $\{u_k\}\in K$ converges to $u$ weakly, how to prove that $u\in K$
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1answer
18 views

Showing that the trace of a positive operator is independent of orthonormal base [on hold]

let $T$ be a positive operator on a separable Hilbert space. let $\{e_n\}$ be an orthonormal base for the space, and suppose the trace of $T$ is finite, i.e. $$tr(T)= \sum_{n=1}^{\infty}(Te_n,e_n) ...
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1answer
24 views

Uniformly Converging Sequence but Not Normly Converging

I want to find a sequence $(f_n) \subset \mathcal L^1(\mathbb R)$ of integrable functions which uniformly converges to $0$ but with $\int f_n \nrightarrow 0$. I came up with the following example: the ...
1
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1answer
30 views

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded?

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded? I don't even know where to start proving or disproving, ...
0
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2answers
40 views

$\frac{\varphi(x+h)-\varphi(x)}{h}\to\varphi'(x)$ uniformly as $h\to 0$?

Let $\varphi$ be a bounded, differentiable function on $\mathbb{R}$ such that $\varphi'$ is bounded and uniformly continuous on $\mathbb{R}$. We want to prove that ...
4
votes
1answer
37 views

$\sigma$-weak topology versus weak operator topology

The reference text for this question is: Pedersen, Analysis Now, GTM 118. The $\sigma$-weak topology on $B(H)$ (the bounded linear operators on a Hilbert space $H$) is the weak$^*$-topology on ...
2
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1answer
24 views

Asymptotic behaviour of Fourier transform: $|F[f]|=|\lambda^{-k}F[f^{(k)}]|$ for absolutely continuous $f$

I read in Kolmogorov-Fomin's (p. 429 here) that if function $f:\mathbb{R}\to\mathbb{C}$ is such that $f^{(k-1)}$ [the $(k-1)$-th order derivative] on any finite interval and if $f,...,f^{(k)}\in ...
6
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0answers
41 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $(\frac{X_i}{n^{\alpha}})_{i=1}^n$? More ...
1
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1answer
29 views

The polar of a set: Importance and Applications

Given a duality $(X,Y)$, for any subset $A\subset X$, we can define the polar set $A^{\circ}\subset Y$. In this sense, the polar relates sets and sets in the dual space. Since cones are sets, we can ...
0
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2answers
24 views

Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...
2
votes
1answer
27 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims $$u = strong - \lim_{\epsilon\to 0} ...
2
votes
1answer
47 views

Weak and weak* topology coincide for a non-reflexive space that is isomorpic to its dual?

There are Banach spaces which are isomorphic to their second dual but not reflexive (most famously, the James space). Now let $X$ be such a space and $X'$ be its dual space and let $\phi:X\to X''$ be ...
2
votes
1answer
19 views

Some closed subspace of $l_2$?

$(a)$ I was trying to define a continuous linear map $T$ on $l_2$ whose kernel would be the $A$ and can conclude $A=T^{-1}(0)$ and hence closed set? could anyone help me to solve any of one?
3
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0answers
78 views

Functional inequalities involving cubing and incrementing

Consider the set $S$ of positive increasing invertible functions $f$ satisfying: $$f((x+1)^3-1)≤(f(x)+1)^3-1$$ $$f(x^3)≥(f(x))^³$$ $$f(x)+1≤f(x+1)$$ for all positive real $x$. Clearly the identity ...
3
votes
1answer
29 views

Ultraweak closed left ideal of a von Neumann algebra

The following is a proposition of Takesaki's Operator Theory: My questions are: 1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ...
2
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0answers
39 views

norm over differentiable functions computable from derivatives only

I'm running an optimization algorithm, minimizing the norm $||f-\hat f||$ of some objective function $f(x_1,x_2,x_3,y_1,y_2,y_3)$. The function $f$ cannot be computed directly, but its second ...
2
votes
1answer
41 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...
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2answers
25 views

The fixed point in Brouwer's Theorem need not be unique.

What does it mean for a fixed point to be unique? I'm thinking that it means that you can have multiple values of a fixed point. But, a fixed point is one where $f(x) = x$. So to have repeated ...
2
votes
1answer
45 views

Infinite direct sum of Hilbert spaces

Let $\{H_i\}_{i \in I}$ be an infinite collection of Hilbert spaces. I am trying to understand their "Hilbert space direct sum". $\bigoplus H_i$ (algebraic sum) is an inner product space in a ...
0
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2answers
39 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
1
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1answer
14 views

Operator which is invariant in terms of left shift

Let $l:l^{\infty}(\mathbb R)\rightarrow \mathbb R$ be a linear map( where $l^{\infty}$ is the set of bounded sequences) such that the following holds: (i) $l(Tx)=l(x)$, where T is the left shift ...
1
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1answer
58 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
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votes
2answers
53 views

C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
0
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1answer
13 views

Proof (?) about weak contractions. Please check to see if I'm going about this correctly?

If $f:M \rightarrow M$ satisfies that $\forall x,y \in M$, if $x≠y$ then $d(f(x),f(y)) < d(x,y)$, then $f$ is a weak contraction. Is a weak contraction a contraction? I saw a counter example on ...