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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If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space.

My professor mentioned this fact in class. FACT: If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space. He mentioned that he had never seen the ...
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29 views

Is $(H_0^1,\|\cdot\|_{L^2})$ a closed subspace of $L^2$?

Let $-\infty<a<b<\infty$ and $f\in L^2(a,b)$. Suppose $(f_n)$ is a sequence in $H_0^1(a,b)$ such that $\|f_n-f\|_{L^2}\overset{n\to\infty}{\longrightarrow}0$. Can we conclude that $f\in ...
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Finite dimension and total boundedness

Let $T:X\to Y$ be a bounded operator between Banach spaces $X$ and $Y$. Assume that for any $\epsilon >0$ there is a finite-dimensional subspace $Y_\epsilon\subset Y$ so that $\|Q_\epsilon ...
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32 views

Continuity of a multivariate function

I'm trying to show that $\langle , \rangle$ is continuous on $V{\times}V$, ($V$ an inner product space). I've tried approaching it by showing $\langle\vec x,\vec y\rangle\rightarrow\langle\vec a,\vec ...
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28 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
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31 views

About dual and quotients

I need help with this exercise, which is supposed to be related with Hahn Banach theorem. It states the following: If $M$ is a closed subspace of a normed space $X$, and defining $M_0=\lbrace{\psi\in ...
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58 views

This is Banach space?

Let me denote by $C_{0}(\mathbb{R})$ the set of continuous functions which tend to zero at + and - $\infty$. I am wondering if it is true that $(C_{0}(\mathbb{R}), \Vert . \Vert_{\infty})$ is a ...
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23 views

Example for a two norms and vector space which are equivalent on this Vectorspace

I need an example where the two norms and the vector space are equivalent on the vector space.
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A question about Hahn-Banach theorem.

Hahn-Banach theorem states: Let $X$ be a real vector space and $p$ a sublinear functional on it. Also, let $Z\subset X$ be a subspace, and $f$ a sublinear functional on $Z$. Also, for all $z\in ...
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31 views

Is this a valid argument to show $|a|^q \in L^{(p/q)'}(\Omega)$?

Brezis' functional analysis book includes a solution to one of its exercises that is looking somewhat fishy to me: while attempting to formalize the last step in the solution, I have arrived at a ...
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30 views

Subspaces of finite dimensional Hilbert spaces

This might be a trivial question but please point out exactly where my reasoning is incorrect. Is every subspace of $\mathbb{R}^n$ closed since $\mathbb{R}^n$ with the dot product is a finite ...
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29 views

On the isomorphism between bounded sesquilinear forms and bounded operators between two Hilbert spaces

Let $H$ and $K$ be two Hilbert spaces. Let $S(H,K)$ be the vector space of bounded sesquilinear forms $u:H\otimes \overline{K}\to\mathbb{C}$, and let $B(H,K)$ be bounded linear operators from $H$ to ...
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49 views

infinite dimensional normed space is not complete

I want to show that space X with norm of sum is not complete. for any x, we have $x= \sum^n_{k=1}c_{k}b_k$ where $b_k\in basis$ and $c_k\in field$ norm is $\lvert\lvert x \rvert\rvert = $ ...
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49 views

Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ...
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79 views

Totally boundedness of a compact operator [closed]

Let $T:\ell_2(\mathbb N)\longrightarrow \ell_2(\mathbb N)$ bounded linear operator such that $$T(\{x_n\})=\{x_n/n\}.$$ I need to prove that $TB(\ell_2(\mathbb N))$, that is closed unit ball in ...
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38 views

Rudin. “Functional Analysis”, Theorem 1.18

If $\Lambda$ is a linear functional on a TVS, which is not necessarily normed, what does the following statement mean: (d) $\Lambda$ is bounded in some neighbourhood $V$ of $0$. For people ...
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33 views

How to find a function in $ L_{2}(0,\infty)$ but not $L_{q}(0,\infty)$, $q\neq 2 $

How to find $x\in L_{2}(0,\infty)$ \ $L_{q}(0,\infty)$ , $q\neq 2 $ i tried $\frac{1}{t.lnt}$ with various degrees on $t$ and $ln(t)$ Could you please help me with this question
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Spectrum of perturbed operator's

Let $G$ be a normal operator with compact resolvent acting on a Hilbert space $H$ such that $\ker G \neq \{0\}$. If $P$ denotes the orthogonal projection onto $\ker G$, and if $\{\lambda_n\}$ are ...
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37 views

If $\lambda$ is isolated in $\sigma(u)$, then $E(\left\{\lambda\right\})(H)=\ker(u-\lambda)$.

This is a Question 2.11 from Murphy's book: C$^*$-algebras and Operator Theory: Let $H$ be a Hilbert space. Let $u\in B(H)$ be a normal operator with spectral resolution of the identity $E$. (a) ...
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32 views

About an application of the open mapping theorem

I need some help proving this result. I know that it has relation with the open mapping theorem, but I can't see how. It states the following: Let $S:E_1\to E_2$ surjective and linear, and ...
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47 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
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39 views

Domains of Lipschitz class are domains of type A.

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. We say that $\Omega$ is of type $A$ if there exists a constant, $A$, such that \begin{equation} |\Omega\cap B_{\rho}(x_0)|\geq A\rho^n ...
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How to get this inequality using induction (analysis)

Consider the following functions $\theta:\Bbb R\to\Bbb R$ and $\Theta:\Bbb R^n\to\Bbb R$, sucha that: $$ \theta(x) := \begin{cases} 1-|x| & \text{if $|x|\le1$} \\ 0 & \text{if $1\le|x|$} \\ ...
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A well known Property of a Mollifier?

Let me denote by $\rho_{n}(x)=n \rho(nx)$ where $\rho$ is any positive smooth compactly supported function (let say in $[-2,2]$) whose integral over $\mathbb R$ is equal to 1. Does anyone see why ...
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35 views

Dense subspace in a Hilbert space

Let $H$ be a Hilbert Space and $\{e_n\}_{n\in\mathbb{N}}$ an orthonormal basis. Now let $(x_n)$ be a sequence in $H$ satisfying $$\sum_{n=1}^{\infty}||x_n-e_n||^2<1.$$ Prove that ...
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44 views

Common orthogonal basis for $L^2$ and $H^1$

How can we obtain a common orthogonal basis for the space $L^2(U)$ and $H^1(U)$ for some bounded open subset of $\mathbb{R}^n$? That this can be done is mentioned in Evans's Partial Differential ...
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18 views

About the boundary of a set of the form $Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$

Let $\Omega$ be a bounded (open) domain. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. ...
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20 views

About $C_c^\infty((0,T)\times \Omega)$

Let $\Omega = \Omega_1 \cup \Omega_2 \cup \Gamma$ where $\Omega_1, \Omega_2$ are open domains in $\mathbb{R}^n$ and $\Gamma$ has measure zero. $\Gamma$ is the interface between $\Omega_1$ and ...
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51 views

Coercivity of bilinear form

I want to show that there is a unique solution for $$-u''=f$$ with boundary condition $$-u'(0)+u(0)=u'(1)=0$$ so I define bilinear form $$a(v,w) = \int\limits_0^1 {v'} w'dx + v(0)w(0)$$ so I should ...
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52 views

Find $B $ s.t. $B=\lbrace y\in s \mid \sum x_ky_k \text{ converges for all }x\in A\rbrace$ for $A=l_\infty$

let $A$ be a sunbset of $s$, and let, $$B=\left\lbrace y\in s \mid \sum x_ky_k\text{ converges for all }x\in A\right\rbrace$$ Could you please find $B$ if (i) $A=s$ and (ii) $A=l_\infty$ we worked ...
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Limit of non-negative non increasing function is Lebesgue integrable

So the problem is: Suppose $g_n \in C[a,b]$ is a sequence of non-negative functions such that $g_n(x)$ is a non-increasing sequence for each $x \in R$. Show that the limiting function $\displaystyle ...
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66 views

Can natural quotient map between Banach spaces be closed?

Let $X$ be a Banach space and $M$ be closed subspace of X, and let $q:X\to X/M$ be the natural quotient map. I know that $q$ is an open map. I wish to find an example of $X$ and $M$ such that $q$ is ...
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18 views

Changing space in the supreme, surjectivity argument

Let $\Omega$ an open subset of $\mathbb{R}^2$ with continuous boundary and $f:X\to L_0^2(\Omega)$ a surjective function, with $X$ another Hilbert space and $\displaystyle L_0^2(\Omega):=\left\{u\in ...
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104 views

Problem from Evans' PDE book, chapter 5, problem 5

I'm taking my first theoretical math course in a year and am bashing my head against a rock with this problem. "The sets $U,V$ are open, with $V \subset \subset U $ (compactly contained). Show that ...
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31 views

Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

Let $D$ be an open subset of $\mathbb{R}^n$ , $p$ and $q$ be in $(1,\infty)$ such that $p^ {-1} +q^ {-1} = 1$. Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$ ...
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26 views

Differentiability in $\mathbb R^n$

Let $U\in \mathbb{R}^n$ be open, and let $f:U\to \mathbb{R}^m$, and let $a\in U$. Let $\|\cdot\|'$ be a norm on $\mathbb{R}^n$, and let $\|\cdot\|''$ be a norm on $\mathbb{R}^m$. Prove that $f$ is ...
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36 views

Show $T$ cannot be a compact operator

Let $(X,\lVert\cdot\Vert_x)$ and $(Y,\lVert\cdot\Vert_y)$ be normed spaces, X be infinite dimensional and $T\in\mathcal{L}(X,Y)$ Which has the property: there exists $m>0$ such that $ \Vert{T ...
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There are $u$ in $W^{1,p}(D)$ and a subsequence $\left\{ u_{m_{k}}\right\} $ such that $\left\{ u_{m_{k}}\right\} $ weakly converges to $u$.

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $p \in (1,\infty)$, and {$u_m$} be a bounded sequence in $W^{1,p}(D)$. Then there are $u$ in $W^{1,p}(D)$ and a subsequence ...
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57 views

Notation Question (Meaning of double inclusion symbols)

What does the notation $\subset \subset$ mean? In my class notes, our prof writes $\Omega \subset \subset \mathbb{R}^{n}$ to mean that "$\Omega$ is a convex subset of $\mathbb{R}^{n}$". Is that all ...
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33 views

Fourier series using other bases?

The theory of Fourier series, representing a reasonable function by an infinite sum of exponential functions, is very well-developed. In addition to basic functional-analytic results there are things ...
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18 views

Pitt's theorem on automatic compactness of bounded operators between sequence spaces

Why is it called Pitt's theorem? I couldn't locate the origin of the statement.
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76 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
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36 views

Characterization of Continuous Linear Transformation

The Question is: Let $V$ & $W$ be two Normed Linear Spaces & let $T: V \to W$ be a linear transformation. Show that: $T$ is continuous iff $T$ maps Cauchy Sequences in $V$ into Cauchy ...
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45 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
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Why do we need the Hahn-Banach Theorem to extend a bounded linear functional?

I'm beginning with functional analysis and I have a related question. It concerns the Hahn-Banach theorem. In particular, I cannot appreciate its value, most probably because I don't understand it. I ...
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66 views

Bounded subsequence in Sobolev Space

The following is an exercise. Let $I=(0,1)$ and let $(u_n)$ be a bounded sequence in Sobolev space $W^{1,p}$, First question: does "bounded" here means that (for a suitable $M$) $$ \| u_n \|_p ...
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18 views

A transformation recipe for functional calculus of a self-adjoint operator?

Consider a self-adjoint operator $\operatorname{A}$ on a Hilbertspace $\mathcal{A}$ and its spectral decomposition according to the spectral theorem: $$A = \int_{\mathbb{R}} \lambda \;dP_\lambda$$ ...
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54 views

$U$ linear and bounded, is an isomorphism $\iff$ $U$ is invertible and $U^{-1}=U^*$

"Let $H$ and $G$ be Hilbert spaces and let $U:H \rightarrow G$ be a bounded operator. Prove that $U$ is an isomorphism $\iff$ $U$ is invertible and $U^{-1}=U^*$." I have denoted $U^*$ to be the ...
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31 views

For bounded linear operators is it true that $(T_1+T_2)x=T_1x+T_2x$?

I wounder if the above is true just from the definition for bounded linear operators, ie by requireing for some linear operator $T$ that $||Tx||\le c||x||$, or if this is a relationship that needs to ...
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64 views

Linear operator with dense range but not full range

Can you give an example of a Banach space $E$ and a linear operator $A \in \mathcal L(E)$ such that $A$ has dense range but not full range, i.e. $ran(A) \neq E = \overline{ran}(A)$? Clearly, $E$ has ...