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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Can I define a bounded sequence whose Banach limit is not unique?

Banach limit, as a non-constructive object, is not unique. The Banach limit for some sequences, say, convergent sequences, sequences satisfying $a_n = a_{n+m}$ for all $n$ and some $m$, the Banach ...
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1answer
28 views

Weak convergence and lim inf and lim sup of the sequence of norms

Assume $x_n$ is a sequence in a Banach space that converges weakly to $x$. Then we know that $\|x\| \leq \lim \inf \|x_n\|$. 1)But can we say that $\lim \inf \|x_n\| < \infty$ or is this in ...
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1answer
19 views

Projection theorem for nonclosed subspaces

Is there a substitute for the projection theorem for Hilbertspaces (if $M$ is a closed subspace of $H$ then $H = M \oplus M^\perp$) in the case that $M$ is a linear subspace of $H$ which is not ...
2
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2answers
54 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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0answers
31 views

Can someone explain this problem I am having with the proof of the Riesz-Fischer theorem

Here is the form of the theorem I have; Let $\{e_n\}_{n=1}^{\infty} \in H$ be an orthonormal set (H a Hilbert space with inner product $(.,.)$) and let $(a_n)_{n=1}^{\infty}$ be an arbitrary sequence ...
2
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1answer
21 views

Natural structure over a set of measurable functions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $U$ be the set of all measurable functions over $(\Omega, \mathcal{F}, \mathbb{P})$ - i.e. the elements of $U$ are all measurable ...
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1answer
27 views

$X$ is inner product space then its completion is Hilbert space?

I have trouble finding a way to prove that the completion of my innerproduct space $X$ is a Hilbert space. How do I know that the norm on the completion of $X$ is induced by an innerproduct? Thanks ...
2
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1answer
21 views

Are the solutions of a Sturm-Liouville equation entire in the spectral parameter?

In $[1]$ the following (paraphrased) claim is made: Let $q\in L^1_{loc}([0,\infty);\mathbb{R})$, and suppose $\varphi$ and $\theta$ solve the one-dimensional Schrödinger equation \begin{equation} ...
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0answers
14 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequality of the Schatten-p (quasi-)norm, ...
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35 views

Prob. 2.7-10 in Kreyszig's Functional Analysis Book: Is my solution good enough for anciliary purposes?

With valuable help from the SE community, I've managed to come up with the following solution to Prob. 10 after Sec. 2.7 in Introductory Functional Analysis With Applications by Erwine Kreyszig. I ...
3
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1answer
46 views

Strongest topology makes unit ball compact

Let $X$ be a Banach space and $X^*$ be its dual. Let $\mathbb{B}^*$ be the closed unit ball in $X^*$. The Banach-Alaoglu Theorem asserts that $\mathbb{B}^*$ is compact in the topology $\sigma(X^*, ...
3
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1answer
42 views

When does $f_n(x) = a_n \times (1 - nx)$ converge uniformly?

The sequence of functions $\{f_n\}_n$ is defined on $[0,1]$ by: $$f_n(x) = a_n \times (1 - nx),\ {\rm\ if}\ x \in ]0,\frac{1}{n}],$$ and $f_n(x) = 0$ otherwise, where $(a_n)_n$ is a positive ...
1
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1answer
33 views

Showing a set is not norm bounded

Consider the set $K = \{x(n) : x(n) \in \ell^p, \sum |x(n)| < 1\}$ $(0 < p < 1)$. I have shown that this set is weakly bounded, but I am now asked to show it is not originally bounded. where ...
2
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1answer
27 views

$k^2 e^{ikx} \rightharpoonup 0$ in the Sense of Distributions

So I'm concerned showing $k^2 e^{ikx} \rightharpoonup 0$ in the sense of distributions or in other words for any $\phi \in C_c^{\infty}(\mathbb{R})$ we have for any $\epsilon > 0$ $$ \left\lvert ...
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1answer
27 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
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0answers
12 views

Tensor norm for matrix algebra over an $L^p$ operator algebra

This is a question about whether a certain tensor norm has a certain property. The setting is that of $L^p$ operator algebras (i.e. norm-closed subalgebras of $L^p(X,\mu)$ for some measure space ...
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0answers
10 views

Selfadjoint Operators: Relative Boundedness

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard an operator: ...
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1answer
41 views

When is the $L^{2}$ norm smaller than the $H^{-1}$ norm?

If $u\in L^{2}$ then we can define the functional: $$u(\phi)=\int \phi u $$ for all $\phi \in H^{1}_{o} $. which means that $u$ is a linear functional in $H^{-1}$. Now for any $f\in H^{-1}$ ...
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0answers
25 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
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votes
0answers
13 views

What are endomorphism, automorphism of an operator algebra (or C*-algebra)?

Are these definitions true? Let $A$ be an operator algebra. Thus: 1) $f:A \to A$ belong to $End(A)\ $ if $\ f\ $ is homomorphism $ \ $i.e. $\ $ $f(ab)=f(a)f(b)\ $ for each $a,b \in A$. 2) $f:A ...
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1answer
96 views

Integral of a square compared to the square of an integral

What can be said about a complex valued, continuous function $f$, defined on $[0,1]$, such that: $$ \int_0^1{|f|^2}=\left|\int_0^1{f}\right|^2 $$ I encountered this form as part of an exercise. ...
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1answer
32 views

Eigenvectors belonging to different eigenvalues are linearly indepenent. Problem with proof.

I am reading up on some functional analysis to prepare for an upcoming subject. I came across the following theorem and proof. Let $X$ be a linear space and $T:X\to X$ be a linear operator. Then the ...
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0answers
11 views

Approximate unit for an ideal and its limit

Let $A$ be a C*-algebra and $I$ is a closed ideal of $A$. If $(\pi, H)$ is a cyclic representation of $A$, Could we show $$\pi(u_i)\xi \to \xi $$ where $\{u_i\}$ is an approximate unit for $I$ and ...
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2answers
30 views

question on the proof of uniqueness of completion

In the book by Erwin Kreyszig, "Introductory Functional Analysis with Applications": I don't understand something in proof of completion theorem. In the fourth part of proof that about uniqueness of ...
3
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0answers
35 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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3answers
36 views

Injectivity of $T:C[0,1]\rightarrow C[0,1]$ where $T(x)(t):=\int_0^t x(s)ds $

Prob. 2.7-9 in Erwin Kreyszig's "Introductory Functional Analysis with Applications": Is this map injective? Let $C[0,1]$ denote the normed space of all (real or complex-valued) functions defined and ...
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0answers
16 views

Subspace of $L^p(X,\Sigma,\lambda)$

Consider $R$-valued functions in $L^p(X,\Sigma,\lambda)$, where $X=X^1\times X^2$, $\Sigma=\Sigma^1\times \Sigma^2$ and $\lambda=\lambda^1\times \lambda^2$ For given $i$, does the subsapce $M=\{f\in ...
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1answer
25 views

Norm closed subspace, and weak* dense

Represent $\ell^1$ as the space of all real functions $x$ on $S= \{(m,n): m\geq 1, n \geq 1\}$, such that $$ \|x\|_1 = \sum |x(m,n)| < \infty. $$ Let $c_0$ be the space of all real functions ...
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0answers
26 views

How can I find the closure of $P[a,b]$ [closed]

Let $P[a,b]$ the space of all polynomials on the interval $[a,b]$ clearly $P[a,b]$ is a subspace of $C[a,b]$ but how can find the closure of $P[a,b]$ , In special case $[0,1]$ .
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0answers
41 views

Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
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0answers
7 views

Transformation of graphs, finding the values of unknowns

I am a second grade IB student using "Mathematics Standard Level for the IB Diploma, Cambridge" book.This is the question I have a problem with: "Let f(x)=(3x-5):(x-2) a) Find the value of constants ...
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1answer
20 views

The linearness of extension of linear bounded operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$G: ...
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2answers
40 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
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0answers
8 views

Decomposing continuous linear functionals on a locally convex space with 2 seminorms

Let $X$ be a locally convex topological vector space whose topology is defined by the seminorms $\rho_1$ and $\rho_2$. (Let us require that topological vector spaces be Hausdorff by definition.) If ...
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0answers
17 views

Diffusion semigroup generated by Laplacian [closed]

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
4
votes
1answer
95 views
+50

How to check some topological concepts in product and direct sum spaces

Given $a=(a_i)_{i=1}^\infty$ with $a_i \geq 0$ and $b=(b_i)_{i=1}^\infty$ with $b_i \in \mathbb{R}$, let $$E_i = \lbrace (x_n)_{n=1}^\infty : n^{b_i}|x_n|\leq a_i, \forall n\in \mathbb{N} \rbrace$$ ...
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0answers
21 views

Commutative Operators from QM

In Theoretical Chemistry, there seems to be a lot of assumptions about mathematics that are incorporated without justification. One example that I found questionable is this: $$\int \Psi_1^*\ ...
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1answer
48 views
2
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0answers
26 views

A linear functional of $H^k$

Define $H^k$ as usual: $$H^k=\left\{f; \int (1+|\xi|^2)^k|\widehat{f}(\xi)|^2d\xi<\infty\right\}$$ Define $l(u)=\int (1+|\xi|^2)^k\widehat{u}(\xi)d\xi$, how to show that $l\in (H^k)^*$. I tried ...
1
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1answer
30 views

What's the spectrum of the operator $g\longmapsto f\cdot g$?

What is the spectrum of the operator: $$T: C[0, 1]\longrightarrow C[0, 1], g\longmapsto f\cdot g$$ where $f\in C[0, 1]$ is a fixed function? Here I'm considering the space $C[0, 1]$ endowed with the ...
2
votes
1answer
26 views

Prove operator is isometry

Let $(X,\mathcal{A},m,T)$ be a probability preserving transformation. Prove that the operator $U:f\mapsto f\circ T$ satisfies $$ \|Uf\|_{p}=\|f\|_{p} $$ for every $1\le p<\infty$. My idea: $$ ...
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1answer
51 views

Is this set of $C[0,1]$ compact?

Let $x_1, x_2 \in C[0,1]$; $x_1(t)<x_2(t)$ $\forall t \in [0,1]$; $$ M=\{x \in C[0,1]: x_1(t) \leq x(t) \leq x_2(t) \, \forall t \in [0,1]\}. $$ Is $M$ compact or not?
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1answer
53 views

Smoothness of inverse Fourier transform

Let $\hat{f}(\xi)$ be a smooth function on $\mathbb{R}^n$ that decays like $|D^\alpha_\xi \hat{f}(\xi)| \lesssim (1 + |\xi|^2)^{-\frac{1}{4}(1 + |\alpha|)}$, where $\alpha$ is a multi-index such that ...
0
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0answers
53 views

Compactness result

I try to prove this lemma: Let $\mathrm{H}_{\text{comp}}^1(\Bbb R^{\mathrm{N}})$ be the subspace of $\mathrm{H}^1(\Bbb R^{\mathrm{N}})$ of functions with compact support. For each ...
3
votes
0answers
51 views

Prove that: $\{Kf_n \ ; n \in \mathbb N\}$ is equicontinuous

Let $a, b \in \mathbb R$. Let: $K \in C([a,b]^2)$. $(f_n)_{n \ge 1}$ be a bounded sequence of functions in $C([a,b])$ equipped with $||\cdot||_{\infty}$ the norm of uniform convergence. $Kf: [a,b] ...
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0answers
14 views

An example for generalized operator projection.

Let be generalized operator$\Pi_{C}:X\longrightarrow C$ and $\Pi_{C}(x)=\hat{x} \quad \hat{x}:\varphi(x,\hat{x})=inf\varphi(x,y)$\ that $y\in C$ NOW To give an example , high definition
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0answers
17 views

A question on the well-posedness of of p-Laplacian

Can it be shown that the problem \begin{eqnarray} -\Delta_{p} u &=& f(u),\nonumber\\ u|_{\partial\Omega} &=& g, \end{eqnarray} well-posed similar to the case when $p=2$?.
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0answers
21 views

connections between polar set, polar cone?

Given a set $S$, its polar cone http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone and its polar set http://en.wikipedia.org/wiki/Polar_set are defined. Could some please tell me the ...
1
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1answer
27 views

For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$. Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev ...
1
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0answers
16 views

Hahn Banach theorem and supporting hyperplane theorem

The question is out of Rudin Functional analysis Chapter 3 problem 1. Call a set $H \subset \mathbb{R}$ a hyperplane if there exists real numbers $a_1,\ldots, a_n, c$ (with $a_i \neq 0$ for at least ...