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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Examples of applications of mononotone and pseudomonotone operators

Hi I am aware that the following question is quite broad, but I would appreciate any feedback even if it is in the form a reference. I am interested in some standard examples in engineering (or any ...
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18 views

Geometric Intuition behind the Dual Norm?

What is the geometric intuition behind the dual norm, $\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$ Specifically, if possible in terms of hyperplanes defined by $x$ and $z$. My interest ...
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19 views

$L_2$ norm and RHKS norm in Hilbert spaces $\mathcal{H}$ [on hold]

There is a claim: A function $f\in \mathcal{H} \text{ (i.e., Hilbert space)}$ if and only if $\|f\|_{\mathcal{H}}:=\left \langle f,f \right \rangle_{\mathcal{H}} < \infty$, and its $L_2$ norm is ...
4
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0answers
29 views

Check proof about range of bounded linear operator.

I have to prove that the range $\mathcal{R}(T)$ of bounded linear operator $T:X\rightarrow Y$; $X,Y$ normed spaces need not be closed in $Y$. As a hint I'm given that I could consider ...
3
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2answers
39 views

Demonstration of $\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= \int_0^a f(x) \,dx$

Good morning, Can you give me a help to demonstrate this proposition: $f$ is an even and continuous function on the interval $[-a,a], a>0$. Demonstrate: $$\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= ...
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0answers
49 views

Spectrum: Continuous?

Declaration Declare convergence of sets pointwise: $S_n\to S:\iff\chi_{S_n}\to\chi_S$ Problem Given a Banach algebra with unit $1\in\mathcal{A}$. Consider a sequence: $A_n\to A$. Then do the ...
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2answers
49 views

How can I calculate the norm of this linear functional on $\mathbb R^3$

I've been trying to calculate the norm of $\phi\colon \mathbb R^3\rightarrow \mathbb R$ defined by $\phi(x,y,z)=1.2x-0.6y$. I really don't know how to this. I did note that $\phi(-2y,y,z)=-3y$, ...
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1answer
62 views

Continuous Nowhere Differentiable Function [on hold]

Define a function $\,f:\mathbb{R}\rightarrow \mathbb{R}_{+}$ by: $$ f(x)=\left|x-2\,\left \lfloor \frac{x+1}{2}\right \rfloor \right|. $$ Here are some known properties about the function $f$: ...
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24 views

how to prove the sum of projections is also a projection [on hold]

Let $P$ and $Q$ be projections. Show $P+Q$ is a projection iff $ranP\perp ranQ$. If $P+Q$ is a projection ,then $ran(P+Q)=ranP +ranQ$ and $ker(P+Q)=kerP\cap kerQ√$
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0answers
43 views

Prove that $SO_2(\mathbb {C})$ is unbounded by providing a sequence of such that norm is unbounded. [on hold]

Can someone help proving that $SO_2(\mathbb {C})$ is unbounded. A simple answer by providing a sequence whose norm is unbounded will be appreciated.
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2answers
29 views

Limit of a sequence problem

Suppose that $(x_n)$ is a sequence such that $\lim_{n\to\infty}\sum_{k=1}^n\frac{{x^4_k}}n=0.$ How do I show that $\lim_{n\to\infty}\sum_{k=1}^n\frac{{x_k}}n=0$?
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0answers
25 views

about ultrapowers

Let $\mathcal{R}$ denote the hyperfinite type $II_{1}$ factor, with $\mathcal{R}^{\omega}$ the ultrapower of $\mathcal{R}$, in respect to some ultrafilter $\omega$ on $\mathbb{N}$. I'm reading a paper ...
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2answers
39 views

Does equicontinuity imply uniform continuity?

If $\{f_n(x)\}$ is an equicontinuous family of functions, does it follow that each function is uniformly continuous? I am a bit confused since in the Arzela-Ascoli theorem, equicontinuity is seems to ...
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1answer
33 views

Image of a commutative C*-algebra

Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism ...
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3answers
58 views

Sequence and series problem

How do I show that the sequence $(x_n)$ defined by $$x_ {n+1} = \left(1-\frac{1}{n}\right) ^2 x_n + \frac{1}{n}, \forall \,n \in \Bbb{N}-\left\{0\right\}$$ converges? and to what limit?
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15 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, ...
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1answer
30 views

stuff involving adjoint, self adjoint [on hold]

Let $T: V \to V$ be a linear transformation relative to a finite dimensional Euclidean space $V$ (real or complex). Prove that there exists linear transformation $T^*: V \to V$ (called the adjoint ...
2
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1answer
52 views

Solve Integral equation with convolution

I have to solve the following integral equation \begin{align*} \int_{-\infty}^\infty e^{-y^2} \log \left( \int_{-\infty}^\infty e^{-(y-x-t)^2} f(t) dt\right) dy=-cx^2 \end{align*} where $c$ is some ...
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2answers
82 views

One dimensional vector space and not Hausdorff

I read that all vector spaces that do not have the Hausdorff property and are one-dimensional need to have the trivial topology. I am not quite sure how to approach this problem, but I would like to ...
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19 views

Confirmation of redundant term in equation

Hi I have a quick simple question from a proof I am working through. I just want confirmation. Are the $[x,y]$ outside of the weak closure redundant in equation (2) of the attachment, since ...
2
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0answers
56 views

Functional derivatives on manifolds

This might be more of a physics question, but it is mathematics-related, I hope I am not out of place with this. Let $(M,\mathcal{S},g)$ be a smooth, $n$-dimensional manifold equipped with a Riemann ...
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1answer
72 views

$p^2=p$ in closure of ideal $I$ of Banach algebra implies $p\in I$.

Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$. Show: $p\in I$. Can someone give me a little hint how to solve this, please?
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17 views

If a bidual space $X^{**}$ is isomorphic to a dual space $Y^*$, is there any relation between $(X, \rho|_X)^*$ and $Y$?

If a bidual space $X^{**}$ is isomorphically renormed to a dual space $Y^*$, in other words there exists a norm $\rho$ s.t.$(X^{**},\rho )=Y^*$, then is there any relation between $(X, \rho|_X)^*$ ...
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1answer
18 views

The Spectral Radius of a Product of Two Hilbert-Space Operators

I’m given a Hilbert space $ \mathcal{H} $ such that $ \dim(\mathcal{H}) > 1 $, and I’m supposed to construct two operators $ A $ and $ B $ on $ \mathcal{H} $ such that $ r(A B) \neq r(A) r(B) $. Is ...
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1answer
24 views

Growth conditions for partial differential equations

Hi I am interested in what the exact purpose is of growth conditions associated with solving partial differential equations. For example the following pde: $$\text{div}(a(x,u,\nabla u)) + c(c,u,\nabla ...
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1answer
34 views

representation for Banach algebra [on hold]

How we can represent any Banach algebra as a subspace or Subalgebra of Cb(X)?( in isometrically isomorphic concept) Cb(X)= the set of all complex-valued, bounded , continuous functions on X
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17 views

Embedding in Generated C*-Algebra

Given C*-algebras $\mathcal{A}$ and $\mathcal{A}'$. Suppose they have common elements: $$\mathcal{A}\cap\mathcal{A}'\neq\varnothing$$ Then is there a generated C*-algebra containing both and is it ...
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39 views

Intersection of C*-Algebras again C*-Algebra

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{A}'$. Is their intersection necessarily a C*-algebra again? So I started like this: $$A,B\in\mathcal{A}\cap\mathcal{A}'\implies ...
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consider the normed linear spaces $(\mathcal C[0,1], ||.|| _i)$.what can you conclude about the correspoding open unit balls?

consider the normed linear spaces $$(\mathcal C[0,1], ||.|| _1), \;(\mathcal C[0,1], ||.|| _2),\;(\mathcal C[0,1], ||.|| _3)\ldots, (\mathcal C[0,1], ||.|| _p)$$ and $(\mathcal C[0,1], ||.|| ...
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2answers
59 views

Is it true that a continuous function with compact support is uniformly continuous?

I've been trying to prove the given $f:\mathbb R\rightarrow \mathbb C$ continuous with compact support, $f$ is uniformly continuous. I don't know if it's true or not, but it is highly plausible and ...
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1answer
35 views

Nonunital C*-Algebras: Morphism contractive?

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Suppose it misses a unit $1\notin\mathcal{A}$. Consider a *-morphism $\pi:\mathcal{A}\to\mathcal{B}$. Then it is contractive: ...
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1answer
50 views

Check if a function is L2

I want to check if a function $f$ defined on $[0,T]$ is a $L_2$ function. What I know is $f$ is a $L_1$ function. (but $f$ could be not bounded) So I want to use an inequality like $$ ...
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2answers
42 views

Is $C^1[a,b]$ a Banach space as a subspace of $C[a,b]$?

Let $C[a,b]$ be the space of continuous functions on $[a,b]$ with the norm $$ \left\Vert{f}\right\Vert=\max_{a \leq t \leq b}\left| f(t)\right| $$ Then $C[a,b]$ is a Banach space. Let's view ...
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63 views

Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to approximate this ...
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4answers
143 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
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1answer
25 views

Is the image of a $*$-homomorphism $\pi:\mathcal{A}\to\mathcal{B}$ closed if $\pi(1)\neq 1$?

Setting Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with unit $1\in\mathcal{A}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$ without $\pi[1]=1\in\mathcal{B}$. Especially, it is a ...
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0answers
21 views

What is the dual of $L^{\infty}(K)$ with K a compact subset of $R^n$?

I know it's probably hard to describe the dual of $L^{\infty}(X)$ for a general $X$. But can we describe it when $X$ is just a compact subset of a Euclidean space?
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0answers
47 views

Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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0answers
15 views

Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...
3
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1answer
51 views

For elements of the intersection of C*-algebras, can the spectra be distinct depending on the algebra?

Problem Given unital C*-algebras $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard an element $A\in\mathcal{A}\cap\mathcal{A}'$. Can it happen that: ...
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1answer
35 views

Show that a subspace is closed in Hilbert space $H$

Let $u\in B(H)$ , $\lambda < 0$. Also we have $\|(u-\lambda)x\|\geq |\lambda|\|x\|$. So $u-\lambda$ is bounded below. To show $(u-\lambda)(H)$ is closed in $H$, suppose $\{(u-\lambda)x_n\}$ be ...
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+50

Sobolev space on $M \times (0,\infty)$, $M$ compact closed manifold

I want to know things like definitions of Sobolev spaces on a manifold of the form $M \times (0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times (0,\infty)$ is a ...
2
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1answer
65 views

Associated Legendre polynomials

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
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1answer
44 views

Open mapping lemma - are these versions equivalent?

Here is a version the Open Mapping Lemma given in class : Let $X$ be a Banach space and $Y$ be a normed space. Let $T : X\rightarrow Y$ be a bounded linear map. Assume there exist $M \geq 0$ and ...
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1answer
32 views

Topological Spaces: Pre-Uniform Structures

Disclaimer This thread is meant to record. See: Answer own Question Reference It is a follow-up to: Uniform Spaces: Neighborhood System It has relevance to: TVS: Uniform Structure Problem Given ...
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1answer
74 views
+50

Spectrum of left shift operator: take two

This is my second attempt at calculating the spectrum of the left shift operator on a Hilbert space. I got stuck again and I would be grateful if someone could help. (You can find my previous (failed) ...
3
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0answers
17 views

Dual of $l^p$ Direct sum

I am asked to show that the $l^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $l^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
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1answer
44 views

Prove that $T^*$ is injective iff $ImT$ Is dense

Let X,Y be two normed spaces, and $T:X\rightarrow Y$ a bounded linear operator. prove that the adjoint operator $T^*$ ($T^*f(x)=f(Tx)$ is injective iff $ImT$ is dense any help would be great guys. I ...
2
votes
1answer
63 views

Proving existence of a linear functional

Let $(X, \| \cdot \|)$ be a normed space, and let $A, B ⊂ X$ be disjoint convex sets such that $B$ is closed and $A$ is compact. Prove that there exists $\varphi ∈ X^*$ such that $$\sup_{a\in A} ...
3
votes
1answer
42 views

Show that an operator is well-defined

Let $v\in B(H)$, Define $u:|v|H\to H$ such that $u(|v|\xi) = v\xi$ . To show the map $u$ is well-defined, the author writes $$\||v|\xi\|^2=\langle v^*v\xi,\xi\rangle = \|v\xi\|^2$$ But I do not know ...