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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57 views

Infinite series of integrals of $L^2$ functions

I'm hoping someone can help me with this integration problem I've been struggling with. Let $\{f_n\}$ be a sequence in $L^2(\mathbb{R})$ such that $\sum_{n=1}^\infty \lVert f_n\rVert^2_2<\infty$ ...
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1answer
77 views

Showing the compactness of a limit operator.

I was trying to solve this exercise from Kreyszig's book, section 8.1 exercise number 10. My attempt was try to show that the operators in the sequence are bounded, but I don't find it. If this fact ...
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0answers
30 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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1answer
37 views

Show $l^p$ embeds in $L^p(0,1)$

Let $l^p$ be the standard sequence space indexed by $\mathbb N$. I've heard it claimed that $l^p$ embeds into $L^p(0,1)$ in such a way that $$L^p(0,1)=l^p\oplus S$$ for some closed subspace $S\subset ...
2
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1answer
33 views

$C^1(\bar \Omega)$ is a Banach space

My professor gave a proof of the completeness of $(C^1(\bar \Omega),\|\cdot \|_{C^1})$ based on the fundamental theorem of calculus. I though about an alternative and I would like to know whether this ...
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1answer
66 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
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1answer
28 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...
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1answer
14 views

correspondence between linear functional and function

Any Schwartz or $L^p$ function $g$ can be identified with a linear functional via which way? $T_g(f)=\int gf$ or $T_g(f)=\int g\bar f$ ? I have seen these two different definitions in different ...
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0answers
24 views

A question on the Banach Contraction Mapping Principle

The BCMP states that in a complete metric space $X$, a contraction mapping $T$ on $X$ has a unique fixed point, i.e. if $T$ satisfies $d(Tx, Ty) \le k d(x,y)$ such that $0 \le k < 1$, then $T$ has ...
0
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1answer
17 views

$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$

I took a passage from a textbook regarding equivalent conditions of having an orthonormal sequence in a Hilbert space H. Why is the equality $$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle ...
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2answers
43 views

schauder basis for $\ell_\infty$ [duplicate]

I know that $\ell_\infty$ is not separable, therefore has no Schauder basis. However I cannot understand why the set $\{e_1, e_2, e_3, \dotsc \}$ where $e_1=(1,0,0,\dotsc), e_2=(0,1,0,0,\dotsc), ...
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1answer
47 views

Example of a non-convex set for which A + A = 2A

Give an example of a non-convex subset $A$ of a (real / complex) vector space $V$ for which $$A + A = 2A$$ Here the sum / multiplication with a scalar of a subset is defined in the obvious way. I ...
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2answers
30 views

Hahn Banach Theorem: Clarification on meaning of extending a functional?

Hahn Banach Theorem: Given linear (vector) space $\mathbb{X}$, define $u \in \mathbb{L} \subset \mathbb{X}$, $A,B,C$ functionals, A sublinear. $A:\mathbb{L} \to \mathbb{R}, B:\mathbb{L} \to ...
2
votes
1answer
28 views

Partial Isometries: Final

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By the C*-property: $$J=JJ^*J\iff P^2=P=P^*$$ Note that in any ...
2
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0answers
41 views

Construct an operator that fixes the equivalence class of Cauchy sequences

Let $X$ be a Banach space and $\overline{X}$ be its unique completion. We know that $\overline{X}$ can be partitioned into equivalence classes of Cauchy sequences via the relation $\sim$: $$ \{x_n\} ...
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0answers
14 views

An analytical expression for the degree of a map from the sphere to itself

Good morning to everyone. I have found in this paper the following statement (not verbatim): "Let $\phi$ be a smooth map from $\mathbb{R}^2$ to the $3$-dimensional sphere $S^2$ which is constant far ...
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1answer
48 views

A nice Application of Baire category Theorem.

Let $f_n$ be continuous functions on a complete metric space $X$ such that $f_n(x)> 0$ for every $x \in X$. Let $A ={x \in X \mid \liminf f_n(x) =0 }$. Prove that $A$ is a countable intersection ...
2
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0answers
91 views

Is “almost all function” a well defined concept?

I am working on a problem which has well defined properties for the vast majority of all PDFs. I would like to make a quantitative statement along the lines of "for almost all distributions, P holds". ...
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1answer
76 views

What is the difference between an function and functional?

Can someone give an example that would point out the difference between a function and a functional in a very simple way?
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2answers
64 views

Does anybody know the definition of $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$, where $0<\alpha<1$?

I hope someone can give me the definition of the following: $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$ for some $0<\alpha<1$ and some domain $\Omega$. In this context they also talk about ...
2
votes
1answer
25 views

(Operator) norm inequality for continuous functions

Let $f,g$ be two non-negative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t,$ $\forall t\in[0,\infty)$. Let be $A$ be a bounded linear operator acting on a Hilbert space. Then I was ...
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0answers
20 views

Does $\chi_{A_\epsilon} \to \chi_{\{x : f(x) = 0\}}$ if $A_\epsilon = \{ x \in \Omega : 0 \leq f(x) < \epsilon\}$?

Define $A_\epsilon = \{ x \in \Omega : 0 \leq f(x) < \epsilon\}$ where $f$ is a given function say in $L^1(\Omega)$. Is it true that $$\chi_{A_\epsilon} \to \chi_{\{x : f(x) = 0\}}$$ pointwise? ...
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0answers
23 views

Variation problem : Euler equation using direct method

Consider a map $I : C^1[0,1]\rightarrow \mathbb{R}$ defined by $$I (f)=\int_0^1\frac{1}{2}(f'(x))^2 -V(f(x))dx$$ where $V(t)\leq 0$ for all $t \in \mathbb{R}$ is smooth. Let $\Gamma=\{f\in C^1[0,1]|\| ...
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1answer
12 views

If $u_n \rightharpoonup^* u$ in $L^\infty$, does $\int u_n^+ \to \int u^+$?

Let $\Omega$ be bounded. Suppose that $u_n \rightharpoonup^* u$ in $L^\infty(\Omega)$ and $u_n \to u$ in $H^{-1/2}(\Omega)$ (that is negative a half, not a typo). Does this somehow imply that ...
2
votes
2answers
69 views

Are all matrices linear operators?

Given $A \in \mathbb{K}^{n\times m}$ a matrix, can we think of $A$ as an operator? In what context do matrices satisfy the definition of operator?
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1answer
26 views

a nontrivial inequality in the proof of weak solution of biharmonic equation

Hi I am looking at the post discussed about weak solution of biharmonic equation Proving unique weak solution. I am having trouble verifying statement 2: The bilinear operator is coercive, The claim ...
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0answers
46 views

Quotient spaces - $\Bbb R^1\hookrightarrow \Bbb R^3$

I am trying to understand quotient spaces, and I constructed my own example to do this: $(\Bbb R=\{(a,0,0)|a\in \Bbb R^1\}) \hookrightarrow (\Bbb R^3=\{(\alpha,\beta,\gamma)|\alpha,\beta,\gamma \in ...
2
votes
1answer
77 views

Intuitive functional analysis book

I want to know a functional analysis book like Terence tao's real analysis and measure theory book, full of intuition. I am aware of linear algebra, real analysis, measure theory, Probability theory.
2
votes
1answer
55 views

Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space ...
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0answers
48 views

Is functional analysis related to or used in algebraic geometry in any way?

I'm curious about whether there's a link (and, no, this question was not motivated by the fact that Grothendieck used to be a functional analyst!) between these two subjects. Are the techniques from ...
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1answer
41 views

Need help proving $n(T)=n(T^*)$ for finite dimensions.

In my book this is showed: Let H and K be complex Hilbert spaces and let $T\in B(H,K)$. There exists a unique operator $T^* \in B(K,H)$ such that $(Tx,y)=(x,T^*y)$ for all $x\in H$ ...
2
votes
1answer
52 views

A very simple question: what spaces of function does the laplace transform map from and into?

Given a function $f$, we can write $f:\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the laplace transform operator ...
2
votes
1answer
73 views

Sobolev spaces over closed domains.

I am currently working through books on Sobolev spaces and I notice that these spaces are almost always defined over open domains, i.e. we look at $W^{m,p}(\Omega)$, where $\Omega$ is open. Because ...
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2answers
31 views

Show that function is in L^2

I'm going through a paper and I came across the following statement: Given $\mathbf{q}_h \in \mathbf{V}_h(\Omega)$ we have to show that $\nabla\cdot\mathbf{q}_h$ is well defined and in ...
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1answer
78 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
3
votes
1answer
69 views

concept of the classification of $C^\ast$-algebras, introduction/overview

I don't have a specific mathematical problem at the moment but nevertheless I hope, my question is suitable for math.stackexchange. I'm interested in $C^\ast$-algebras and I would like to begin with ...
1
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1answer
22 views

A question on Operator of a Banach Space

For any $x \in X$ where $X$ is a Banach space, is there a non-trivial bounded operator $T \in B(X)$ such that $T(x)=x$? I mean is there any way to verify the existence of such an operator for any $x ...
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votes
0answers
30 views

Equivalence of Norms and Open Mapping Theorem

Let $V$ be a vector space with two norms $||\quad||_{1}$, $||\quad||_{2}$, making $V$ a complete normed vector space. Assume $\exists C$ (constant) such that: $||v||_{2} \leq C||v||_{1}, \forall v ...
3
votes
1answer
40 views

Norm of an integral operator $L^1 \rightarrow L^\infty$

Let $T:L^1(\mathbb{R}^n)\rightarrow L^\infty(\mathbb{R}^n)$ be an integral operator, i.e. there exist $K:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that for all $f\in ...
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1answer
42 views

How do you quickly show whether an operator is compact?

In my text on functional analysis, the author defines an operator on normed space as compact if it: continuous transforms bounded sets into relatively compact sets Okay, number 1 we can work with. ...
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1answer
40 views

If $\dim X=n$ then for any norm in $X$, $X$ is complete. [duplicate]

I know there are standard proofs for this theorem, but I need to prove it by contradiction or proving that $\dim X=\infty$. I thought maybe using Hahn-Banach? Thanks.
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0answers
17 views

Potential operator

I have this operator $A:H\rightarrow H$ where $H$ is a Hilbert space, defined by: $$Au(t)=\int_0^1k(t,s) f(s,u(s)) ds$$ I want to find conditions on $f:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$ ...
3
votes
1answer
59 views

Solution to Equation $Ax=f$ in Hilbert Space

Question. Let $H$ be a separable Hilbert space with complete orthonormal basis $\left\{u_{k}\right\}_{k=1}^{\infty}$, let $H_{n}:=\text{span}\left\{u_{1},\ldots,u_{n}\right\}$, and let ...
3
votes
2answers
190 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...
0
votes
1answer
28 views

extension by zero for sobolev functions.

Given some function in $H^1(A)$, and if $B$ is an open subset (A also open) containing $A$, do we get an element of $H^1(B)$ if we just extend by 0? I dont think so, but what would be a simlple ...
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2answers
26 views

Definition of associative algebra over a field

In the definition of an algebra over a field in the wiki entry , it states that an algebra over a field is a vector space equipped with a bilinear product. Question: Does anyone know how a bilinear ...
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1answer
52 views

How to understand this notation on Fourier transformation?

For a function $f$, recall the Fourier trnasformation $\widehat{f}(u)=\int_R e^{iux}f(x)dx$ (Maybe someone call it Fourier inversion, but it doesn't matter). Now let $T$ be a bounded self-adjoint ...
2
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0answers
28 views

Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
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0answers
19 views

Understanding function space associated with vector space

In a vector space, pick a vector $v=(v(1), v(2),...).$ In function space, pick a function $f=2x$ in which $f=\{f(r)\}_{r\in R},\,f$ is considered as a point in function space as the case in vector ...
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1answer
17 views

Getting the unique element in the Riesz-Frechet Theorem.

I have this thorem in my book, H', denotes the dual space, that is the set of bounded linear operators from X to the field over X. The way they got the unique element seems very interesting. Does ...