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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44 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 ...
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1answer
25 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 ...
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1answer
52 views

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

Let $E$ is a closed bounded set of $\mathbb{R}$. Is $C(E$ a dual of any linear norm space?
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1answer
229 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 ...
2
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1answer
101 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 ...
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1answer
44 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 $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
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1answer
19 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 ...
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1answer
75 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 ...
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65 views

Is the martingale property 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 ...
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58 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: ...
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1answer
103 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 ...
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1answer
43 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 $C(K,X)$ ...
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1answer
74 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 ...
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25 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!
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1answer
122 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?
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1answer
38 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 ...
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1answer
31 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
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0answers
93 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 ...
2
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0answers
427 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}\| & = ...
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1answer
35 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 ...
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1answer
70 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 ...
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1answer
92 views

Complexification the real inner product space

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, ...
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2answers
41 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, ...
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1answer
97 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 ...
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1answer
213 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, ...
3
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1answer
216 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 ...
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2answers
145 views

Isometry <=> Adjoint left inverse [duplicate]

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

Isometric <=> Left Inverse Adjoint

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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1answer
99 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.
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33 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
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1answer
247 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 ...
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0answers
85 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 ...
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1answer
39 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 ...
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3answers
90 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 ...
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0answers
113 views

Mean value of a function over the n-sphere's surface.

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 ...
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2answers
64 views

Linear Projections: Bounded/Continuous?

Are linear (nonorthogonal) projections on (pre) Hilbert spaces necessarily bounded/continuous? (can you give a proof or counterexample)
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0answers
39 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?
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1answer
63 views

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

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
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0answers
38 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 $a_n x_n$ ...
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1answer
47 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
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1answer
112 views

Does the closed unit ball of $C(E)$ have no extreme points?

Let $E$ be a bounded closed set in $\mathbb R^n$. Does the closed unit ball of $C(E)$ (the space of continuous functions on $E$ with supremum norm) have no extreme points?
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66 views

German and French word for Basic sequence

Wikipedia says that a sequence $(x_n)$ is a basic sequence iff it is a Schauder basis of its closed linear span. I was wondering whether there is a French or German word for 'basic sequence'? How ...
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1answer
58 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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1answer
21 views

Is $\phi(h)=\displaystyle \sup_{x\in X}{\|h-x\|}$ continuous on Hilbert space when $X$ is bounded?

Let $H$ be a hilbert space, and let $X \subset H$ be a bounded subset of $H$. Let define the function $\phi:H \to H$ by the rule $\phi(h)=\displaystyle \sup_{x\in X}{\|h-x\|}$. I want to know if this ...
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1answer
22 views

About a proposition of Willem's book

Let $\Omega$ be an open subset of $\mathbb{R}^N$ and let $2<p<\infty$. The functionals $\Psi(u)= \int_{\Omega} |u|^p , \chi (u)= \int_{\Omega} |u^+|^p$ are of class $C^2(L^p(\Omega), ...
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37 views

Proof that these are Fourier coefficients

I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$ In Murphy's book ...
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1answer
606 views

First resolvent equation

$(B,||\cdot||)$: banach space A family $(G_{\alpha})_{\alpha>0}$ of linear operators on $B$ with $D(G_{\alpha})=B$ for all $\alpha>0$ is called a strongly continuous contraction resolvent if ...
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1answer
61 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
4
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2answers
106 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
2
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1answer
83 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...