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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Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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a question about finding the points where Df(x) (derivative of f) is an isomorphism.

Let E be the four-dimensional real vector space $M_{2\times 2}$ of real 2$\times$2 matrices. Show that by setting f(X)=X^2 for 2$\times$2 matix X,we define a continously differentiable function f ...
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9 views

A weak* dense subset intersected with norm ball contains no ball

I'm struggling with this problem in general. Represent $\ell^1$ as the space of all real functions $x$ on $S = \{(m,n): m\geq 1, n \geq 1\}$, such that $$ \|x\|_1 = \sum |x(m,n)| < \infty $$ ...
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8 views

$A\subseteq B(X, Y)$ compact if and only if closed and $Ax$ is conditionally compact

This comes from Exercise 2 of Chapter VI in Dunford & Schwartz. I am trying to prove the following statement: A set $A\subseteq \mathscr{B}(X, Y)$ is compact in the strong operator topology if ...
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11 views

Compact Operators: Adjoint

Given Banach spaces $E$ and $F$. Consider a bounded operator: $$T:E\to F:\quad\|T\|<\infty$$ As a result due to Schauder: $$T\text{ compact}\iff T'\text{ compact}$$ How to prove this fact?
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Closed Convex Sets of $L^1([0,1])$ and Minimal Norm

Let $M$ be the set of all $f\in L^1([0,1])$, relative to Lebesgue measure, such that $$\int_0^1f(t)\,dt=1.$$ Show that $M$ is a closed convex subset of $L^1([0,1])$ which contains infinitely many ...
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32 views

Confused on notation Rudin Functional analysis

The question is stated as follows Represent $\ell^1$ as the space of all real functions $x$ on $S= \{(m,n): m\geq 1, n \geq 1\}$, such that $$ \|x\|_1 = \sum |x(m,n)| < \infty. $$ Let $c_0$ be ...
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1answer
20 views

Banach space of p-Lipschitz functions

Given $p\in\mathbb{R}$, consider the space: $$ Lip(p) = \left\{f:[0,1] \longrightarrow \mathbb{R} : \mbox{ $f$ is $p$-Lipschitz} \right\}$$ i.e.: there is $M>0$ such that ...
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1answer
33 views

A lower bound for $\left|\int_a^b f(t) dt\right|$.

It is known that, if $f : [a; b] \rightarrow \mathbb R$ is integrable, then $f$ is bounded, $|f|$ is integrable and $$\left|\int_a^b f(t) dt\right|\leq \int_a^b|f(t)|dt$$ My question is the following. ...
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2answers
41 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
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12 views

Continuity of functional in L^{3/2} [0,2]

Let $A : L^{3/2} [0,2] \rightarrow \mathbb{C}$ ; $f\mapsto A f = \int_0^1 e^x f(x)$ I need to prove that this functional is continuous, so I have to show the linearity of the functional, and also ...
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9 views

Proof involving strong continuous semigroups

Let $T(t)$ be a $C_{0}$ semigroup on the Hilbert space $X$ with infinitesimal generator $A$ and let $\rho\in(0,1)$. I want to prove that $\displaystyle \sup_{t\ge 0}||T(t)-I||\le \rho$ is equivalent ...
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22 views

Compact Operators: Trace

Problem Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard orthonormal bases: ...
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10 views

Basis for functions with particular fall-off behaviour

Consider the space of functions on $\mathbb{R}$ that fall off like $x^n$, when expanded around infinity. Is it possible to construct a basis in this function space?
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26 views

What is a good definition of a weak solution?

I am interested in any heuristic or formal necessary and sufficient conditions that a good definition of a weak solution of a PDE problem must satisfy. This question is motivated by trying to show ...
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1answer
24 views

Showing that A is NOT an infinitesimal generator

As a state space, choose $X=L^{2}(0,1)$. Let $A$ be defined as $\displaystyle Af=\frac{df}{d\zeta}$ with domain $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous and $\frac{df}{d\zeta}\in ...
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11 views

Does $\partial_b u(\cdot,b) \in L^2(A)$ for fixed $b$ imply that $u(\cdot,b) \in L^2(A)$?

Let $u$ be defined on $A \times B$ where $A$ and $B$ are two bounded domains, and write $u=u(a,b)$. Suppose that the weak derivative $\partial_b u(\cdot,b) \in L^2(A)$ for fixed $b$. Does this imply ...
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38 views

completeness proof

I'm looking at this exercise solution and there is a last step which I do not really understand. Consider the set of continuous functions on the interval $X$, that is $C(X):=􏰁\{f:X→R \mid f \ ...
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1answer
19 views

Quick question on countably dense subsets of Hilbert spaces.

Hi I am reading a proof in my functional analysis notes and there is a step I don't really understand; Since H (infinite dimensional Hilbert space) is separable it contains a countable sense subset ...
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2answers
13 views

Finding the infinitesimal generation of a strongly continuous semigroup

Let $X$ be a Hilbert space, $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I have already shown that $T(t)$ defines a $C_{0}$ semigroup. But now I need ...
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Need help for construct DLSQ-spline in $B$-form

In Schumaker, Larry L. "Computing bivariate splines in scattered data fitting and the finite-element method." Numerical Algorithms 48.1-3 (2008): 237-260 (Link to journal, link to author page for ...
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49 views

Verify that $f $ is constant

Let $K(x) = x$ on $[-1,1]$ and extend period to real space. Suppose $f \in L^2[-1,1]$ satisfies $\int_{-1}^{1} K(x-t)f(t) dt = 0$ for any $x$. Then, $f$ is a constant function. I have no idea for ...
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1answer
21 views

Meager Set in Function Space (Constructing Sequence of Functions)

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^1[0,1], f \textrm{ strictly increasing} \}$ equipped with the topology of uniform convergence. Consider the subset $A =\{ f \in X \; | \; ...
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4answers
935 views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
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16 views

operator on a fixed element in Banach algebra

Let $X$ be a Banach lattice algebra endowed with an ordering $\leq$. $T=\{T(t)\}_{t\geq 0}$ be the positive semigroup defined on $X$. $F: X_+\rightarrow X$ is a continuous mapping such that ...
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18 views

Integration over Haar Measure

How to perform the following integration $I = \int_{O(n)}\int_{O(m)}f(\mathbf{H}_1,\mathbf{X},\mathbf{H}_2,\mathbf{Y})d\mathbf{H}_1d\mathbf{H}_2$ ; $\hspace{10pt}m\le n$ where, $d\mathbf{H}_1$ ...
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1answer
21 views

Bounded operator from $H^1$ to $L^2$

I am working through the nice Finite Element notes by Douglas N. Arnold at the moment. They can be found here. A teeny tiny detail in the proof of Lemma 7.4 gives me a headache and I am looking for ...
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Small operators in Hahn fields

Let $H=K((G))$ be a Hahn field over $K$ with monomial group $G$. Let $D$ be a K-vector subspace of $K((G))$ such that if $(a_i)_{i\in I}$ ($I$ possibly infinite) is a sequence in $D$ such that ...
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1answer
26 views

Mackey topology on $X'$

Does the Mackey topology $\tau(X',X)$ coincide with the operator norm topology on the dual $X'$ of a normed space $X$?
3
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1answer
38 views

Poincare-like inequality

I would like to prove that there exists $C>0$ such that $$\| u \|^2_{L^2(B(0,1))} \leq C \left ( \| \nabla u \|^2_{L^2(B(0,1))} + \| u \|^2_{L^2(\partial B(0,1))} \right )$$ for every $u \in ...
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1answer
83 views

If $u\in H^2(R^n)$, how to prove $\|D^2u\|_{L^2}$ is equal to $\|\Delta u\|_{L^2}$ using Fourier transforms?

Problem: If $u\in H^2(R^n)$, how to prove $\|D^2u\|_{L^2}$ is equal to $\|\Delta u\|_{L^2}$ using Fourier transforms? My first question: Is it right to prove this using integration by parts as ...
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3answers
55 views

What is abstraction of direction in considering vectors such as used in Engineering & Physics?

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction ...
3
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1answer
28 views

is the complexification of a finitely strictly singular operator itself FSS?

Let $X$ and $Y$ be real Banach spaces, and let $X_\mathbb{C}$ and $Y_\mathbb{C}$ denote their respective complexifications. Suppose $T:X\to Y$ is a bounded linear operator which is finitely strictly ...
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1answer
30 views

Tempered distributions

Let P be a vector whose componentes are polynomials in $\mathbb{R}^n$ and harmonics. its true that exists a polinomial T that $\nabla T = P$? I think this has something to be with fourie transform, ...
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2answers
30 views

Find the spectrum of the operator $T: \ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined by $(Tx)_n = \frac{x_n}{n}$

Consider the linear operator $T:\ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined as $$ (Tx)_n = \frac{x_n}{n}, \quad x \in \ell^2(\mathbb{C}). $$ I can show that it is bounded with norm $\|T\|=1$, ...
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Applications of Banach-Alaoglu theorem in the theory of distributions

Are there some interesting applications of Banach-Alaoglu theorem in the theory of distributions? The theorem provides compact subsets in the $w^*$-topology, so distributions seem a great place for ...
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1answer
40 views

How to show this is a metric?

$d_1(x_1,y_1)$ and $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about ...
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1answer
21 views

wot limit of a sequence of projections

Let $\{P_i\}$ be a net of projections on a Hilbert space , then we can show wot limit of this net is a projection, too. I saw below example of a sequence of projections which its wot limit is not a ...
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2answers
89 views

Ball and addition

I have a this problem I'm trying prove, but I don't understand the addition notion in it. $A$ is a closed subspace of a normed space. Prove that if $x\in A+B(0,\beta)$ for all $\beta>0$ then $x\in ...
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27 views

Minkowski integral inequality in Banach space

Let $X$ be a Banach space of all measurable functions on $\mathbb{R}^d$ with the property: for any non-negative increasing sequence $\{f_n\}\subset X$, we have $\|\lim_{n\to \infty} f_n\|_X=\lim_{n\to ...
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2answers
23 views

Figuring domain of constant $a$ in a equation with some condition

Here is what questions says Question: If $a\in \mathrm{R}$ and the equation $-3(x-[x])^2+2(x-[x])+a^2=0$ (where $[\cdot]$ denotes the greatest integer $\leq x$) has no integral solutions, then all ...
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1answer
49 views

Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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Difference between Cauchy, Weierstrass Completeness and Cantor Completeness principles

What is the most significant difference between Cauchy's principle and Weierstrass Completeness/Cantor's Completeness principles?
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45 views

If every functional of $f$ is smooth, is $f$ smooth?

Let E be a (real or complex) Banach space and suppose $f: \mathbb{R}^n \rightarrow E$ has the property that $\lambda \circ f$ is $C^\infty$ for every bounded linear functional $\lambda \in E^\ast$. ...
4
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1answer
38 views

Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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1answer
29 views

Upper bound for norm of matrix (cf. Example 2.7-7 in Erwine Kreyszig's book)

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the norm space of all ordered $n$-tuples of real numbers with the norm defined as ...
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2answers
22 views

Spectrum of a bilateral shift

Let $u$ be a bilateral shift on Hilbert space $\ell^2(\Bbb Z)$. As unilateral shifts, spectrum $u$ does not contain any eigenvalue. Also $u$ is unitary, so $\sigma(u) \subset \Bbb T$ ($\Bbb T$ means ...
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1answer
34 views

Wot convergence and sot convergence

Let $\{A_n\} $ be a sequence of bounded linear operators on Hilbert space $H$ and $\langle A_n\xi,\eta \rangle \to \langle A \xi,\eta\rangle$ for $\xi,\eta\in H$ with $\|\eta\|=1$. Show that $\|A_n\xi ...
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24 views

Quotient space isometrically isomorphic to $c$

Can some one help me fill the details in the next proof? It will be much better if someone knows a simpler way to do it. The problem states Take $C[0,1]$, with the usual norm $\|\cdot\|_\infty$, ...
2
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1answer
40 views

Real Analysis Exercise

Let $X = \lbrace f \in l^1, \sum_{1}^{\infty} n|f(n)| < \infty\rbrace$. Show that $X$ is a proper dense subspace of $l^1$, hence is not complete. I figured out that $X$ is a proper set of $l^1$, ...