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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Find the value of a such that F(a) achieves its minimum value

Find the value of a such that $F(a)$ achieves its minimum value. $$F(a)=\int_{0}^{\pi/2} \left|\sin x - a\cos x \right| dx $$ I'm trying to use following fact to solve the problem but then I need ...
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25 views

Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

I came across this inequality today: $$\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$$ I realise if we let $h \to 0$ we obtain the derivative on the left hand side so I can see it has ...
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37 views

weak convergence and weak * convergence criterium

I have to solve the following problem Let $X$ be a Banach space. Prove that $x_n\rightharpoonup x$ in $X$ if and only if $\sup||x_n||<+\infty$ there exists a dense subset $E'$ of ...
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18 views

How to represent the function of variables?

I have a function as $$E=\int_\Omega -\log\big( p_i(x)\big) dx$$ where $p_i(x)$ is density distribution which estimated by Parzen window method. $p_i(x)=\frac{1}{\Omega_i} ...
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17 views

Find the point spectrum

Let $x$ and $y$ be nonzero vectors in a Hilbert Space $H$. Let $f(z) = \langle z, x \rangle y$. Find the point spectrum of $f$. I know that $\lambda$ is in the point spectrum if $f(z)=\lambda z$ for ...
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31 views

How to prove a normal operator is reducible?

I want to prove: given a bounded normal operator $T$ on a Hilbert space $H$. If $H_1$ is a $T$-invariant subspace, then the orthogonal complement $W_1$ of $H_1$ is also $T$-invariant. I can prove the ...
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2answers
57 views

Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces ...
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11 views

Non-symmetric case, can we get the minimization problem in Lax Milgram therom?

In the Dietrich braess's book 'Finite Elements'. The Lax-Milgram thm is stated under the condition that $a$ is symmetric(There are many book which also state property of symmetric when getting a ...
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18 views

Aproximation in $W^{1,p}(U)$ with U disconnected.

Consider $U=(-1,0)\cup(0,1)$. Define $$v(x)=\left\{\begin{array}{rc} 0,&\mbox{se}\quad -1<x<0,\\1, &\mbox{se}\quad 0<x<1. \end{array}\right. $$ Clearly $v\in W^{1,p}(U)$ for each ...
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32 views

Is the solution to the heat equation always $C^k$, no matter what the boundary condition is?

Let $\Omega$ be a bounded Lipschitz domain (can be smooth if necessary). Consider the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ $$\text{some Robin boundary condition (B) for $u$}$$ We know ...
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30 views

Finite Dimensional Hilbert Space

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x ...
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0answers
34 views

Lebesgue measurable integration, density

Let $\mathbb{T}$ be the unit circle and $\lambda$ be the Lebesgue measure on $\mathbb{T}$. Let $A_n := e^{2\pi i[1/2^{2n},1/2^{2n+1}]}$, $n\ge 1$. Define a function $f$ on the set of all the Lebesgue ...
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23 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where ...
2
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23 views

Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ is not connected. Prove that $A$ contains a nontrivial idempotent $z$.

While trying to solve the exercise below, I came up with a wrong conclusion, but I can't see why it's wrong. Also I'm accepting suggestions to get the right solution. This is the problem 17 from ...
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1answer
43 views

Continuity of the operator “inverse of” in the space of linear, bounded and bijective operators

Let $(X,\|\ \|_X),(Y,\|\ \|_Y)$ be two Banach spaces over $K$. Definition: $\qquad\qquad\qquad\quad I(X,Y):=\big\{A\in \mathscr B(X,Y):A\text{ is invertible and }A^{-1}\in\mathscr ...
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24 views

Bounds on $L^2$ and $L^{\infty}$ norms in terms of $H^1$-seminorms for functions attaining a zero in a domain.

We have the following result in one dimension: If $f\in C^1([a,b])$ attains a zero in $[a,b]$, then $$| f |_2 \leq (b-a) | f' |_2$$ and $$ | f |_{\infty} \leq (b-a)^{1/2} | f' |_2$$ Is there a ...
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2answers
49 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
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1answer
40 views

Are generic smooth functions analytic?

It is well-known that generic continuous functions are differentiable almost nowhere. I was somewhat surprised to learn in my functional analysis course that the same is true of $\alpha$-Holder ...
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41 views

Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
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1answer
28 views

Definition of inner product space

In the definition, we defined linearity in the first argument, Hermitian symmetry. And these two imply anti-linearity in second argument. Is it equivalent, if I cancel the Hermitian symmetry and only ...
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1answer
28 views

Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C_b[0,1]$

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
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1answer
25 views

If $(U,〈\;⋅\;,\;⋅\;〉),H$ are Hilbert spaces, $W\in U$, $Y\in H$, $Z\in L(U, H)$ and $f\in L(H,L(H,\mathbb R))$, then $〈Y,fZW〉=〈ZW,fY〉$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be Hilbert spaces $W\in U$, $Y\in H$ and $Z\in\mathfrak L(U, H)$$^1$ $f\in\mathfrak L\left(H,\mathfrak L\left(H,\mathbb R\right)\right)$ How ...
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2answers
37 views

Understanding the proof: a linear functional is continuous if and only if it is bounded.

Let $X$ be a normed space. Prove that a linear functional $f:X \to \mathbb{R}$ is continuous if and only if there is a number $ c \in {0, \infty}$ such that $$|f(x)| \leq c||x||$$ for all $x \in X$ ...
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1answer
31 views

Hahn Banach Theorem Application

I want to proof that exists $f \in l_\infty '$ with $f(x) = \lim x_n, \forall x = (x_n) \in c$ and $f(x_1, x_2, x_3,...) = f(x_2,x_3,x_4,...)$ What I have been doing until now: Consider the ...
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1answer
22 views

Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f'' $ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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24 views

Proof of uniqueness of a fixed point

The Banach fixed point theorem: Let $(X,d)$ be a complete metric space and $$f: X \to X$$ be a contractive map then there is a unique fixed point. I am trying to prove the uniqueness of a fixed ...
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32 views

Adjoint of a bounded linear operator is bounded

Suppose $X$ and $Y$ are normed spaces over $\mathbb{R}$ and suppose $T: X \rightarrow Y$ is a bounded linear map. I want to prove that the adjoint map $T^\star : Y^\star \rightarrow X^\star$ is ...
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2answers
26 views

Let $C$ be a convex closed nonempty subset of a Hilbert space $H$. Show that there is a unique element in $C$ with the minimum norm.

Let $H$ be a Hilbert space and $C \subset H $ be a convex, closed and nonempty subset of $H$. Prove that there exists a unique $x_0\in C$ with minimum norm among the elements of $C$. I don't know ...
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1answer
36 views

About a particular linear map between sequence spaces

Let $x \in \ell^1$ and $z \in \ell^2$ taking values in $\mathbb{R}$ and define a linear map $T_z: \ell^1 \rightarrow \ell^2$ as follows: $y_1=0$ and $y_n=\sum_{k=1}^{n-1}z_{n-k}x_k$ for $n\geq 2$. ...
1
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1answer
46 views

Weakly convergent series

Consider sequence $\{x_n\}_{n = 0}^\infty $ in complex Banach space X. Suppose that for all $z \in \mathbb{C}$ such that $|z| < 1$ the series $\sum_{n = 0}^\infty z^nx_n$ is weakly convergent. How ...
2
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2answers
29 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
2
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1answer
48 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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1answer
22 views

How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
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23 views

Convolution of a distribution with a $C^{N}$ function.

I've been working on the following Problem from Friedlander's introduction to the theory of distributions: Show that if $u$ is a distribution of order $N$ and with compact support on ...
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1answer
22 views

Closed, convex and balanced subset of a vector space

Let $(X,\|\ \|)$ be a vector space over $K$ and let $B\subseteq X$ be closed, convex and balanced. I want to prove the following: If $x_0\in X\setminus B\Rightarrow\exists\;f\in X^*$ s.t. ...
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1answer
24 views

Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ ...
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1answer
61 views

Evaluate for $t\in \mathbb{R}$ $\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx$

Evaluate for $t\in \mathbb{R}$ $$\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx.$$ Here is what I have done: Let $f(z)={e^{itz}\over (1+z^2)^2}$. This has two poles $z=i$ $z=-i$ and an essential ...
3
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1answer
84 views

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions?

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions? My attempt to prove this: For contradiction, suppose $X$ is an ...
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1answer
49 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any ...
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26 views

Proving that these functionals are bounded, and finding their norms.

Proving that these functionals are bounded, and finding their norms. $$a.)f_1(x):c_o \to \mathbb R , f_1(x)=\sum_{n=1}^{\infty}\frac{x_n}{2^{n-1}} \\ b.)f_2(x):l_1 \to \mathbb R , ...
1
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1answer
27 views

Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra ...
4
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1answer
48 views

Proving that a set of functions is a linear subspace of a vector space

I am attempting to solve the following problem: Let $V$ be the vector space of all continuous functions $f : R → R$ with point-wise addition and scalar multiplication defined. (a) Show that $M_1$ = ...
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1answer
39 views

Sequence in hilbert space, mutually orthogonal vectors

Let $y_1,y_2,\cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1,\cdots,y_n\}$. Assume that $||y_{n+1}||\leq ||y-y_{n+1}||$ for all $y\in V_n$ for $n=1,2,3,\cdots$. Show ...
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0answers
25 views

Banach space and invertible linear operator

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
2
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1answer
36 views

If $H$ is a Hilbert space, are we able to identify the derivative ${\rm D}f(x)$ at some $x\in H$ of a differentiable $f\in H'$ with an element of $H$?

I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not: Let ...
2
votes
1answer
13 views

Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take ...
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0answers
22 views

Can we derive a Taylor formula for real-valued Fréchet differentiable functions on a normed space?

Using the Lagrange form for the remainder, Taylor's theorem can be stated as follows: Let $I\subseteq\mathbb R$ be an interval, $f\in C^{n+1}(I)$ for some $n\in\mathbb N_0$ and $s,t\in I$ ...
1
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1answer
14 views

positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...
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0answers
15 views

Following integral is in the domai of the operator or not.

Let $\{T(t):t\geq 0\}$ be a $C_0$ semi group on a Banach space $X$ and let $A:D(A)\to X$ be its infinitesimal generator. We know that for $x\in X$, $\int_{0}^{t}T(s)xds\in D(A).$ Can we conclude ...
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0answers
27 views

Need hint to show that operater is compact

Let H be a separable Hilbert space with the basis $\{e_n\}$. If A is an operator defined by $$Ae=\frac{1}{n} e_n$$. Then show that A is compact. I Just need Hint how to solve such type of problem.