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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434 views

Uniform convergence and interchange or sum and integral in Cauchy integral formula

I am working with the Cauchy Integral Formula for a matrix $A$ over a closed contour $C$. I have the following calculation, I believe this is correct, but I don't understand why I am allowed to ...
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1answer
242 views

Riesz Representation theorem-pde

Consider $\sum_{i,j=1}^n \displaystyle\int_{\mathbb{R}^n} \dfrac{\partial^2 u}{\partial^2 x_i} \overline{\dfrac{\partial^2 v}{\partial^2 x_j} } dx + \lambda \displaystyle\int_{\mathbb{R}^n} u ...
2
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1answer
125 views

Is this estimation of an integral right?

Let $f\in\mathscr{S}(\mathbb{R}^3)$ (the Schwartz space), $k>0$ and consider the integral $$\varphi(k)=\int_{\mathbb{R}^3}\frac{e^{ik|x|}}{|x|}f(x)dx$$Is the estimation ...
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0answers
127 views

best approximation property of finite dimensional banach space.

recently i am stuck in a question whose partly answer is known to me. But i want full answer. Can anyone help me? The question is following: Suppose $X$ is a finite dimensional normed linear space ...
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1answer
75 views

About a weak topology on TVS

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
4
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1answer
294 views

Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
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0answers
40 views

“Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
5
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0answers
250 views

Problem of Scottish Book

Does anyone know if the problem 50 to Banach written in The Scottish Book is resolved? The problem is: Prove that the integral of denjoy is a Baire functional in the space M ( that is to say, in ...
7
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1answer
324 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
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1answer
143 views

How to prove $\int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v$ is an inner product in $H^1$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
3
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2answers
213 views

Do $L^2$ convergence and continuity imply pointwise convergence?

It is said here that $L^2$ convergence and continuity imply pointwise convergence (just before paragraph $5.2$) but I can't find how to prove it. Does anyone see how ?
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62 views

A space $X$ that contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and whose dual is not weakly sequentially complete

I want to find an example of a Banach space $X$ which contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and so that $X^*$ is not weakly sequentially complete.
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1answer
107 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
5
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3answers
345 views

How to prove that $(u-v)^+\in W_0^{1,2}(\Omega)$, if $u\in W_0^{1,2}(\Omega)$, $v\geq 0$.

Let $\Omega$ denote a open subset of $\mathbb{R}^n$, and $W^{1,p}(\Omega)$ the Sobolev space of weakly differentiable functions $u\in L^p(\Omega)$ (that is, for which $D_iu$ exists and belongs to ...
2
votes
0answers
85 views

When $\ell^2$-convergence implies $\ell^1$- convergence?

Consider a sequence $(x_n)_{n\in\mathbb N}$ in $\ell^1$ (sequences taking their values in $\mathbb R$), where $x_n=(x_{i,n})_{i\in\mathbb N}$. What are sufficient conditions on the sequence ...
2
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2answers
111 views

When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?

Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth. Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
2
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1answer
73 views

About a subset of $\mathbb{R}^n$: The other direction of Arzela-Ascoli Theorem

Arzela-Ascoli Theorem. Let $(E,d)$ be a compact metric space and denote by $C(E)$ the space of all continuous real valued functions defined on $E$, with respect to the supremum norm $\|\cdot\|_\infty$ ...
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1answer
159 views

linear independence in the dual space

If $V$ is a $N$ dimensional vector space and $l_1,....,l_k$ are linearly independent elements of $V^*$. How to prove that map $V\to R^k$ given by $v\to (l_1(v),...,l_k(v))$ is surjective? It does not ...
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0answers
193 views

for a 'non-reflexive' space, the weak star topology is strictly coarser than the weak topology on the dual space

I'm trying t0 work out the fine details of a claim in "Functional analysis, Sobolev spaces and PDE's" by H. Brezis. Let $J: \begin{cases} E\longrightarrow E^{**} & \\ x \longmapsto J(x) & ...
2
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1answer
76 views

showing to be extreme subset (might use Hahn decomposition Theorem)

I am studying Functional analysis by myself and stumbled this question and am completely at a loss. We want to show that $\{ f \in L^1 [0,1 ] : ||f|| =1 \}$ is an extreme subset of $\{ \mu \in ...
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0answers
102 views

$M+N$ is a closed subspaces of banach space iff $M^{\bot} +N^{\bot}$ is closed subspace of dual

Let $X$ be a Banach space and let $M,N$ be closed subspaces of $X$. I want to prove that $M+N$ is a closed subspace iff $M^{\bot}+N^{\bot}$ is a closed subspace of $X^{\ast}$ (i.e, dual of $X$). Any ...
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2answers
73 views

Computing the Fourier transform of a certain function

Let $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ be the function defined by $f(x)=(1+|x|^2)^{-1}$. Problem: How can I find $\hat{f}$ by direct computation? Remark: This is exercise 8.5 in Rudin's ...
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1k views

Examples of Banach spaces

Which of the following are Banach spaces? A. The set of all real-valued functions $f$, $g$ which are functions of an independent real variable $t$ and are defined and continuous on the closed ...
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1answer
70 views

Why is $\langle f, u \rangle_{H^{-1}, H^1} = (f,u)_{L^2}$ when $f\in L^2 \cap H^1$ and not $\langle f, u \rangle_{H^{-1}, H^1}=(f,u)_{H^1}$?

More generally, if $V \subset H \subset V'$ are Hilbert spaces, why is $$\langle f, u \rangle_{V',V} = (f,u)_{H}$$ when $f\in H \cap V$ and not $$\langle f, u \rangle_{V',V}=(f,u)_{V}?$$ Is this what ...
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1answer
88 views

quasi-inner product problem

Let $X$ a vector normed space on $\mathbb{R}$ and $a:X\times X\rightarrow\mathbb{R}$ such that $a(x,x)\ \geq\ 0,\;\;\; \forall\ x\in X$. $a(x,y)\ =\ {a}(y,x),\;\;\; \forall\ x,\ y\in X$. $a(\alpha x ...
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votes
2answers
134 views

Prove there exist a unique $u\in H^1$ such that $\int_{\Omega}(\kappa\nabla u\cdot\nabla v + \frac{1}{\kappa}uv) =\ \int_{\Omega}fv$ for $f\in L^2$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
8
votes
1answer
88 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
2
votes
2answers
91 views

True or False; Functional Analysis

Given $T: V \to W$ with $V,W$ being Hilbert Spaces. We always have $\| T^ *\| = \| T \|$. I think it is true because of Riesz' Theorem, but I am not sure if a proof is necessary. EDIT: In case ...
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2answers
820 views

Continuous on rationals, discontinuous on irrationals

Let $f: R \rightarrow R$. Show that the set of points of continuity of $f$ is a $G_{\delta}$ set. Explain why it follows from this that there is no function that is continuous on the rationals ...
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3answers
104 views

Convergence in $L^1$

Let $f \in L^1(R)$. Show that $\sum^{\infty}_{n=1} f(x + n)$ converges a.e. Solution: So, ultimately we are going to want $\sum^{\infty}_{n=1} f(x + n) \leq$ something in $L^1$ that converges ...
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0answers
45 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
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0answers
194 views

Application of Stone Weierstrass Theorem for trigonometric polynomials

In the space $C[-\pi, \pi]$ equipped with the sup norm, consider the linear space $M$ spanned by the functions ${ (\cos nx)}_{n\geq0}$ and $ {(\sin nx)}_{n\geq1}$. what is the closure $\bar{M}$ of ...
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1answer
111 views

Separable spaces

Suppose we have a closed separable subspace $A$ of a non separable Hilbert space $H$, then is $A^\perp$ separable. If so what would be a way to prove it, if not why.
2
votes
1answer
194 views

Can a space have both a conditional and an unconditional basis?

Does there exist a Banach space $X$ which admits both a conditional and an unconditional Schauder Basis? If so, can one find an example in the collection of $\ell^p$ spaces? My thoughts so far: ...
2
votes
1answer
109 views

spectrum of two bounded linear operators

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
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1answer
180 views

Bounded linear operator in weak topology

Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
4
votes
1answer
227 views

Normed vector spaces and Banach spaces

Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
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votes
0answers
55 views

Interpretation for the Functional Determinant

Let $S:V \rightarrow V$ be a linear operator on the function space $V$. It is possible to define a functional determinant for $S$ via the zeta function regularization process. In specific we define ...
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votes
1answer
96 views

Fourier transform in $L^2$

I have a function $\phi\in L^2(\mathbb{R}^3)$ and I know that its Fourier transform satisfies the following equation: $$(p^2-A)\hat{\phi}(p)=Q\frac{A+\lambda}{p^2+\lambda}$$ where $Q$ is a constant, ...
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votes
1answer
78 views

Proving function is convex

How do you show that $c + max(0,1-x)^{2}$ is convex where $c$ is a constant? I can graph it and observe that the function is below any line segment between any two points but I am not sure how to ...
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1answer
145 views

About Closed Unit Balls

If $(X, \|\cdot \|_X)$ is a normed vector space over $\mathbb R$, then the closed unit ball of $X$ is given by $$B(X)=\{x\in X: \|x \|_X \le 1\}.$$ If $X^{*}$ is the set of all bounded linear ...
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1answer
98 views

$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. ...
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0answers
201 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
2
votes
2answers
221 views

Examples of truly abstract evolution PDEs?

Let $V \subset H \subset V^*$. Consider the parabolic PDE $$y' = A(t)y + f$$ which is found in many books. Usually under some assumptions on $A(t)$ and $f$, there is a solution $y \in L^2(0,T;V)$ with ...
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0answers
102 views

Inversion formula for Schwartz-space $\mathcal{S}$.

Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
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1answer
121 views

Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?

If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
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vote
1answer
43 views

sobolev spaces on cartesian products

We have a canonical isomorphism $$ C^0(X,C^0(X,Y)) \simeq C^0(X \times Y, Z)$$ given by $f \mapsto \hat{f}$, where $\hat{f}(x,y) = (f(x))(y)$. Is there a similar statement for Sovolev space? For ...
2
votes
1answer
130 views

dot product and relationship to cosine of its angle

In our class, we defined that the term $$ \cos(\alpha)=\frac{\langle x,y \rangle}{\sqrt{\langle x,x\rangle \langle y,y \rangle}}$$ is equal to the cosine of the angle enclosed by $x$ and $y$. ...
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vote
1answer
59 views

Is it true that $C_0^\ast[0,+\infty) = NBV_{loc}[0,+\infty)$

Any function from $BV_{\operatorname{loc}}[0,+\infty)$ defines a continuous linear functional on $C_0[0,+\infty)$. But is it true that any continuous linear functional on $C_0[0,+\infty)$ is given by ...
1
vote
1answer
102 views

How to sample uniformly from an $\epsilon$ ball?

Given a real rectangular matrix $X$, I would like to uniformly sample from the set of real rectangular matrices $\mathbb{M}$ that satisfy $||X-S||\leq \epsilon, \forall S\in\mathbb{M}$ and for a fixed ...