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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3
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1answer
23 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
-1
votes
1answer
20 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
0
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0answers
10 views

The orthogonal projection of $\gamma(x)=2e^{2\pi xi}$ over the subspace generated by …

The orthogonal projection of an element $x_0 \epsilon$ H over a convex set C is the element $y_0 \epsilon$ C such that $\|x_0-y_0\|=\min_{y \in C}\|x_0-y\|$. Find the orthogonal projection of ...
1
vote
0answers
9 views

Laplace transform of bochner integral

$\mathcal{H}$:real hilbert space with inner product $(\cdot,\cdot)$ and norm $||\cdot||:=(\cdot,\cdot)^{1/2}$ A family $(T_{t})_{t>0}$ of linear operators on $\mathcal{H}$ with ...
1
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0answers
23 views

Show that the map $T(x)=x/2+x^{-1}$ is a contraction and find $\alpha$

Let $X=[1,\infty)$. Show that the map $T(x)=x/2+x^{-1}$ is a contraction, and find $\alpha$. Proof: A function $T:X \rightarrow X$ is said to be a contraction if $dist(T(x),T(y)) \leq \alpha ...
2
votes
1answer
18 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
3
votes
1answer
29 views

Is C(E)a dual of any linear norm space?

E is a closed bounded set of R. Is C(E)a dual of any linear norm space?
0
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1answer
27 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
1
vote
1answer
20 views

Banach spaces: Convergence in terms of the Schauder basis.

Let $X$ be a Banach space. Suppose $X$ has a normalized Schauder basis $\{x_n\}_{n \in \Bbb N}$. Let $\{y_n\}_{n \in \Bbb N}$ be a sequence in $X$ converging to $0_X$. For each $n \in \Bbb N$, let ...
0
votes
1answer
18 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
0
votes
1answer
13 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
2
votes
0answers
35 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
0
votes
0answers
24 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
0
votes
0answers
14 views

use Fatou theorem to prouve an convergence

let $u_n$ an sequence uniformaly bounded in $H^1_0(\Omega)$, then, $u_n$ converge weakly to $u$ in $H^1_0$, and strongly in $L^2(\Omega)$ and a.e $x \in \Omega$. Let $g(x,u)$ an Carathedory function ...
0
votes
0answers
12 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
1
vote
0answers
29 views

Open Convex Subsets of Dense Spaces

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following... Suppose the following: ...
0
votes
0answers
6 views

How does invariance of $q$ wrt $\lambda$ for a stationary functional, restrict the function?

Suppose I have the following functional: $$S(q) = \int_{b}^{a}L(t, q(t), q'(t))dt$$ and $q(t) = x(t) + \lambda$, where $\lambda$ is a constant independent of t. If $S(q)$ is stationary for a ...
0
votes
0answers
45 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
0
votes
0answers
24 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
0
votes
0answers
21 views

Why is it true that $-\int_{\Omega}w\partial_t (w^+) = \frac{d}{dt}\int_{\Omega}|w^+|^2$

Why is it true that $$-\int_{\Omega}w(t)\partial_t (w(t)^+) = \frac{d}{dt}\int_{\Omega}|w(t)^+|^2$$ Where the $+$ denotes the positive part of the function (zero otherwise). I get the answer with a ...
1
vote
1answer
13 views

Compact Embeddings of the space of continuous functions

Let $X$ and $Y$ be Banach spaces and assume $\iota:X \hookrightarrow Y$ is a compact linear injection. Let $K$ be a compact topological space (the unit interval if that helps). Then the space $C(K,X)$ ...
0
votes
1answer
30 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
0
votes
0answers
17 views

What is the contour lines of $\frac{2x+y}{2x-y}$?

What is the contour lines of $$f(x)=\frac{2x+y}{2x-y}$$? I need help to describe them... Id like to get help... Thank you!
0
votes
1answer
19 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
1
vote
1answer
30 views

Vector space clarification

I'm asked to decide if the following are vector spaces. A=$\{f:[0,1] \to \mathbb{R}:\int_0^1|f(x)|dx=0$ $\}$ B= $\{f:[0,1] \to \mathbb{R}:f'(x)+4f(x)=0$ and $f(0)=1 $} C=$\{f:[0,1] \to ...
2
votes
1answer
16 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
3
votes
0answers
43 views

Non unit version of Stone-Weierstrass theorem

If we assume the Stone-Weierstrass theorem, how to prove the following statement: If $X$ is compact Hausdorff, $ C(X \to \mathbb R)$ is the set of continuous real valued functions. If $ A \subset ...
1
vote
0answers
44 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
0
votes
1answer
28 views

Continuity of function between Banach space

The question is: Let B be a Banach space and let f from B into B be a linear map such that f^2=f and both Im(f) and Ker(f) are closed. We want to show that f is continuous. So since Ker(f) is closed ...
1
vote
0answers
34 views

Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric

I've been struggling with this problem for the last four hours. The problem is to show that the space of $\mathbb{C}$-valued continuous functions on $[0,1]$ under the metric ...
0
votes
1answer
43 views

complex inner product from the real

Let $V$ be a real inner product space. If $W=V\times V$ with the operations $(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2)$ and $(\alpha +i\beta)(u,v)=(\alpha u-\beta v,\alpha v+\beta u)$, where $u, ...
0
votes
2answers
25 views

Bijective Bounded Operator Extension: Where do the new elements go to?

Given a dense, proper subset of complete spaces: $$X,Y\text{ both complete and }A\subsetneq\overline{A}=X$$ and an operator between them: $$T:A\to Y\text{ continuous, linear and bijective}$$ Now, ...
3
votes
1answer
46 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
0
votes
1answer
28 views

Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
2
votes
0answers
49 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
2
votes
1answer
24 views

Isometry <=> Adjoint left inverse

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse to }T$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle ...
0
votes
0answers
14 views

Isometry: Adjoint = Leftinverse

Given an isometric operator is it true that its adjoint is necessarily leftinverse? My attempt goes like this: $$\langle x,\mathbb{1}\tilde{x}\rangle=\langle x,\tilde{x}\rangle=\langle ...
2
votes
1answer
21 views

Is the dual of a complete topological vector space always complete?

Let $X$ be a complete topological vector space (over $\mathbb{C}$ say), and $X'$ its dual with the weak*-topology. Then is $X'$ always complete? You may assume $X$ is locally convex if you like.
0
votes
0answers
16 views

What is the convergence criterion for linear fixed-point iteration in Banach space?

Consider an iterative process of the form $x^{n+1}=A x^n + b$. When $A$ is a linear operator in $\mathbb R^n$ then the criterion of convergence is $\rho(A)<1$, where $\rho(A)$ is spectral radius of ...
2
votes
0answers
28 views

Chain rule for weak derivatives of $f(u)$ where $f'$ is not bounded but $u$ is?

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$. Suppose $u$ has a weak derivative $u_x$. I want the chain rule $$\partial_x (f(u)) = f'(u)u_x$$ to hold. We know this holds if $f'$ is bounded. But I don't ...
0
votes
0answers
30 views

Riesz Representation Theorem

I am unfamiliar with Quantum Mechanics and all that stuff. I have recently studied Riesz Representation Theorem , I got to know that it justifies ket and the bra notation. Can anyone please give an ...
0
votes
0answers
18 views

Is the function space of a Banach space with product topology a Frechet–Urysohn space?

Consider $E^X$ the space of functions $f: X\to E$ where $X$ is a set and $E$ is a Banach space over $\mathbb{R}$ or $\mathbb{C}$. Using the product topology, then if a sequence of function $f_n \in ...
0
votes
1answer
21 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
0
votes
3answers
71 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
0
votes
0answers
18 views

Mean value of a function over the n-sphere superficie.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
0
votes
0answers
20 views

$\{Q^{(n)}\}$ is tight? [on hold]

Let $Q^{(n)}$ , $n\ge 1$, are probability meaures in $E^{Q_+}$, where $E\subset \mathbb{R}^d$ ($d\ge 1$) is compact, $Q_+=Q\cap [0,\infty )$ and $E^{Q_+}$ denotes the collection of function from $Q_+$ ...
2
votes
2answers
29 views

Linear Projections: Bounded/Continuous?

Are linear (nonorthogonal) projections on (pre) Hilbert spaces necessarily bounded/continuous? (can you give a proof or counterexample)
1
vote
1answer
19 views

Are the continuous linear functions from a norm space to R bounded?

$\{X, \|\cdot\|\}$ a normed space, a function, maping from $X$ to $\mathbb R$, is linear and continuous. Is it a bounded linear function?
0
votes
1answer
50 views

Isometry from $\ell^1$ to $\ell^\infty$

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
1
vote
0answers
22 views

Question about Schauder basis

The question is : Let $B$ be a Banach space and suppose $\{x_n\}$ the Schauder basis and $M$ be the space of sequence of scalars $\{a_n\}$ such that the sup norm of power series of $a_n x_n$ ...