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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20 views

Descomposition on temporal sobolev space

Let $\Omega$ an open subset of $\mathbb{R}^2$ with Lipschitz boundary. Can I descompose in a unique way any $u\in L^2(0,T;L^2(\Omega))$ such that for all $t\in [0,T]$, $u(t)=u_1(t)+u_2(t)$ with ...
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110 views

Partial derivative w.r.t an integration

For example, I have a functional $$J(f)=\int \frac{f(x)}{1+x^2}dx.$$ How to calculate $\frac{\partial J}{\partial f(x)}$? Does it equal to $\int \frac{1}{1+x^2}dx$? It seems that the question is ...
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224 views

Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
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102 views

Show that the following functional is Frechet differentiable in Hilbert space

I need to show that the following functional is Frechet differentiable: $$ f(u) = \|u\|^2_{H} \ \ \text{in a real Hilbert space} \ \ H $$ Solution: As far as I understand, I need to take a Taylor ...
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94 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
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63 views

closed subspace $Y$ implies existence of non-zero linear functional $g$ such that $Y \subset \ker(g)$

I am working on an exercise and I am not sure if I am on the right track, so if anyone could give some hints I would be grateful. The exercise is If $Y$ is a proper closed subspace of $X$, prove that ...
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66 views

Does there exist a function in $L^1$ such that $u * f = f$ for all f in $L^1$

While studying for exams, a practice question came up which is Does there exist a function $u \in L^1({\mathbb{R}^d})$ such that $u * f = f$ for all $f \in L^1({\mathbb{R}^d})$? I was thinking ...
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35 views

An exercise about compact operater

If $A \in \mathfrak{B}(H)$ and $H$ is a Hilbert space, $AT=TA$ for every compact operater $T$, show that $A$ is a multiple of the identity operater. I don't what is "multiple of the identity ...
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308 views

The adjoint of finite rank operater is finite rank

If $T \in \mathfrak{B}_{00}(\mathfrak{H},\mathfrak{K})$, show that $T^{*} \in \mathfrak{B}_{00}(\mathfrak{K},\mathfrak{H})$ and $dim(ran T) = dim(ran T^{*})$. The ...
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88 views

A convex symmetric set in a real vector space is balanced

Show that a convex set in a real vector space is symmetric if and only if the set is balanced. For the backward direction, i.e. if the convex set is balanced in real vector space, then that it is ...
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40 views

Prove that a give sequence of function is a base of $L^2([0,1])$

Consider $(\phi_k)_{k \geq0} \in \mathcal C^{\infty}([0,1])$ with $\phi_k \not\equiv 0 $ such that $$\int_0^1 \phi_k(s) ds = 0, \quad \forall k\geq 1$$ and $$\sup_{ t \in [0,1]} \left | \frac{d}{dt} ...
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126 views

Ideals and exactness of projective tensor product of Banach spaces / algebras

Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to ...
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102 views

Show that the following functional is Frechet differentiable

I am new to this and I need to show that the following functional is Frechet differentiable: \begin{equation} f(u) = \sin(u(1)) \ \ in \ \ C[0,1] \end{equation} What I have already done: ...
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84 views

Complex Power of a differential operator

Let $(X,\|\cdot\|)$ be a Banach space and consider a sequence $B_n \colon X \to X$ of bounded operators. I remember from my course in operator theory that the partial sum $$ S_N = \sum^N_{n = 1} B_n ...
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3answers
99 views

$A^2$ self-adjoint and Compact, prove $A$ has an eigenvalue

Suppose $H$ is a Hilbert space and $A \in L(H)$ is such that $A^2$ is compact and self-adjoint. Prove that $A$ has an eigenvalue. (Here $L(H)$ is the set of bounded linear operators on a Hilbert ...
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102 views

Variants of the bump function.

The title of this question isn't really clear because of the 150 char limit. What I actually want to ask is this: If I would have a bump function for $-1 < x < 1$ and I would have some ...
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66 views

Determinant inequality for trace class operator

Let $A$ be a trace class operator on a Hilbert space. I wonder if there is an estimate of the form $$ |\log \det (I + A)| \le C\|A\|_1, $$ for some constant $C$, where the norm on the right is the ...
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151 views

Using Nemytskii Theorem for Sobolev Spaces

The Nemytskii mappings in Lebesgue spaces theorem is as follows: If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory ...
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55 views

A problem about projective operater

Let $P$ and $Q$ be projective on a Hilbert space $H$. Show that $P+Q$ is projective if and only if $\mbox{ran }P \perp \mbox{ran }Q$. The sufficiency is easy. About the necessity, suppose $P+Q$ is ...
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127 views

Functional Analysis, operator theory, eigenvalues of a operator

We have $$T_\alpha:C[a,b]\to C[a,b]$$ $$T_\alpha f= \alpha f$$ where $C[a,b]=\{ f:[a,b]\to \mathbb{R} \quad f$ is continuous} and $\alpha\in C[a,b]$ fixed. Show: Spectrum of $T_\alpha\equiv ...
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156 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
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80 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
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128 views

Question about projections on Hilbert space

Let $P_i$ be projections from a Hilbert space $\cal{H}$ to its closed subspace $\cal{H}_i$, $i=1,2,\cdots,n$, such that $\sum^n_{i=1} P_i$ is also a projection. And let $P$ be a projection from ...
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1answer
52 views

What is the name of this object?

Suppose I have a convex set $K\subset X$, where $X$ is say a real Hilbert space (for simplicity). Then, given some $a\in \Bbb{R}$, let $$ \hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\} $$ ...
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1answer
32 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
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1answer
50 views

Extension of a zero linear functional

How can I show using Hahn-Banach theorem that, if $E$ is a real vector space, $F$ is a proper vector subspace of $E$, and f is the zero linear functional $f:F\to\mathbb{R}$ such that $f(x)=0$ $\forall ...
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137 views

Is $C[a,b]$ a closed linear subspace of $L^{p}(a,b)$

I am not sure about the last step of my proof: $(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). ...
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94 views

Find the norm of $A$ where $(Af)(t)=tf(t)$

I have the following problem that I would like to ask you about: I have $X$ as my normed linear vector space and $B(X,X)=B(X)$ as my space of all operators $A: X \to X$, where for all $A \in B(X)$ is ...
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64 views

Confusion over an example of weak limit in L2

I understand the definition of weak limit, but I'm confused about the weak $L^2$ limit of the sequence $g_n = n1_{[0,1/n^2]}$. The $L^2$-norm of each of these functions is $1$. Does this mean the ...
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85 views

Linear dimension of banach spaces

Let $X$ be some vector space (over $\mathbb{C}$). Note that if $X$ is of finite dimension we can identify $X$ with $\mathbb{C}^n$ for some natural $n$ and endow it with a norm $||x||=|x_1|+...+|x_n|$. ...
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57 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
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1answer
58 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
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335 views

Hahn–Banach Theorem for Normed Spaces: not unique extension

Let $\ell^{\infty}$ be the set of bounded sequences in $\mathbb{F}$, with the supremum norm. $c \subset \ell^{\infty}$ the sequences whose limit exists. Then there exists a $f \in ...
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1answer
151 views

Arzela-Ascoli net question

Let $X$ be a compact metric space. Let $C(X)$ denote the space of real-valued continuous functions on $X$. A commonly given corollary to the Arzela-Ascoli theorem is: Proposition: If $f_n$ is an ...
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147 views

Topology of pointwise convergence - open sets

Let $X$ be the vector space of all complex functions on $[0,1]$, topologized by the family of seminorms $p_{x}(f)=|f(x)|$, $0\le x\le1$. This topology is called the topology of pointwise convergence. ...
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1answer
49 views

Question about domains of unbounded operators

This is a part of a theorem in Rudin's Functional Analysis, in the chapter on unbounded operators. Let $\mathcal M$ be a $\sigma$-algebra in a set $\Omega$, $H$, a Hilbert space and $E:\mathcal ...
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33 views

The space $\mathcal{D}((0,T);V)$ and its norm/embeddings?

Let $V$ be a Hilbert space. Define $\mathcal{D}((0,T);V)$ to be the set of functions $u:(0,T) \to V$ such $u$ is compactly supported on $(0,T)$ and is a $C^\infty$ test function. What is the norm ...
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41 views

discontinuity of an operator

I want to show that if $X=Y$ is the subspace of $L^1(0,1)$ over $\mathbb C $ consisting of all polynomials, then $T:X \times Y \rightarrow \mathbb C$ given by $T(f,g)=\int_0^1 f(t)g(t) dt$ is not ...
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51 views

$f=g$ in $L^1$ confusion

Let $f$ and $g$ be in $L^1$. Suppose I am given $$f=g\quad\text{as equality in $L^1$.}$$ Then $f-g = 0$, i.e. $\int |f(x)-g(x)| = 0$ so $f=g$ almost everywhere. But $f=g$ also means $\int |f(x)| = ...
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128 views

Second order quasilinear PDE

Some quick question about PDE's. Only recently started studying PDE's so this might be trivial. The second-order quasilinear elliptic equation is given by: $ -\sum_{i=1}^{n} \frac{\partial}{\partial ...
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101 views

Product rule for Banach space-valued differentiable functions?

Let $\Omega \subset \mathbb{R}^n$ be a bounded open set and let $f(\cdot,\cdot)$ and $g(\cdot,\cdot)$ be functions from $[0,T]\times \Omega$ into $\mathbb{R}$. Suppose that $f \in ...
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22 views

Some basic questions about $C^k(I \times A)$

Let $I=[0,T]$ and $A$ be a bounded set that may or may not be compact. Let $f=f(x,t) \in C^k(I \times A)$. Am I right: 1) $f_x, f_t \in C^{k-1}(I \times A)$ 2) $f_{xt}, f_{tx}, f_{xx}, f_{tt} \in ...
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37 views

Showing a function is continuous on a compact set

Let $\Omega$ be bounded and open in $\mathbb{R}^n$. Let $f \in C^1([0,T]\times \Omega).$ If $f(t, \cdot) \in C^1(\overline{\Omega})$ for each $t$, how do I show that $f \in C^1([0,T]\times ...
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1answer
26 views

definition of the set $H(x,y)$

In this book, under the section $1.3$ (The Mazur-Ulam theorem), there are some definitions in which I am not clear. Suppose $X$ is a normed linear space. Let $x,y \in X$. Define the set ...
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49 views

Norms inducing non discrete Hausdorff topology

We know that any norm defined on a vector space V induces a non discrete Hausdorff Topology on V. Is the converse true?
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71 views

determinant identity for invertible finite rank operators

I am currently reading a paper where the following identity, valid for an invertible finite - rank operator $T \colon \mathscr{H} \to \mathscr{H}$ on a separable Hilbert space, is given: $$ \log \det ...
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1answer
74 views

Quotient norm banach space

I need some help in the follow question The question is about the quotient norm: "True or false: The infimum in the definition of the quotient norm: || [x]|| = inf m∈M ||x − m|| is always ...
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1answer
49 views

Representation in Banach space and norms 'induced' by representation

By $G$ we denote some compact group, $X$ stands for some Banach space. Suppose $\pi\colon G\longrightarrow \mathrm{GL}(X)$ to be some representation in $X$. I'm trying to prove that there is an ...
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59 views

The space $C^1([0,T]\times \Omega)$ for $\Omega$ open and bounded

Let $\Omega$ be open and bounded. Is there anything nice I can say about the space $C^1([0,T]\times \Omega)$ and its inclusion in some Bochner like spaces? If $f \in C^1([0,T]\times \Omega)$ then ...
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1answer
59 views

Finite range operator is compact

This theorem is from Rudin book which he says that obvious, but I'm quite confused how to prove it completely. Hope someone can help me clarify. Let $X$, $Y$ be Banach spaces, If $T \in ...