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
votes
1answer
198 views

Extreme points in the set of positive linear functionals of norm $\leq 1$

Let $A$ be a C* algebra, and $S$ the set of positive linear functionals on $A$ in the unit ball of $A^*$ (Which has the weak-* topology.) I am having difficulty seeing that all nonzero extreme points ...
0
votes
1answer
262 views

Entire functions representable in power series

How to prove that an entire function f, which is representable in power series with at least one coefficient is 0, is a polynomial?
2
votes
2answers
581 views

Frechet derivative

I want to find the Fréchet derivative of the following functional: $$ \begin{align} F : C[-1,1] &\rightarrow \mathbb{R}\\ x &\mapsto x(0)\int_0^1 \sin\ x(t) \, dt. \end{align} $$ How can I do ...
2
votes
1answer
261 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $$ \Vert f\Vert_1=\sup_{t\in [0,1]}|f'(t)|$$ Is it complete?
2
votes
3answers
236 views

Balanced but not convex?

In a topological vector space $X$, a subset $S$ is convex if \begin{equation}tS+(1-t)S\subset S\end{equation} for all $t\in (0,1)$. $S$ is balanced if \begin{equation}\alpha S\subset S\end{equation} ...
0
votes
0answers
126 views

Double Dual of $ \ell^\infty$

For my quetion in MO is $\forall X$, $X^{**}$=X$\oplus Y$ for a $Y$ another set I am not really sure in Thomas answer why the first assumption saying that such a $Y$ exist iff the sequence $0 \to X ...
1
vote
1answer
70 views

How can it be proven that the set of all states is a convex set for the W*- algebra?

How can it be proven that the set of all states is a convex set for the W*- algebra? Is the set of all states non-empty?
6
votes
1answer
547 views

Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
1
vote
0answers
112 views

Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...
2
votes
0answers
123 views

About continuous linear functional on the topology generated by linear functionals

I am self-studying P. Lax's functional analysis book. Here is an exercise in p123. it is supposed to be very easy, but I really couldn't see it. Could you anyone help me out? Thanks. Let $\{ ...
3
votes
1answer
147 views

The definition of the weak topology(projective topology) in Kelley-Namioka 's book

In Kelley-Namioka "Linear Topological Spaces",Page 31, it gives two equivalent definitions of projective topology(i.e.,weak topology). I don't know why they are equivalent. The point is I don't kown ...
0
votes
1answer
118 views

Perturbation of Discontinuous function: Outer Semicontinuous Mapping?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a locally bounded, discontinuous, function and let $\delta: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function. Define the ...
2
votes
2answers
73 views

About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$

Suppose $M$ is map from vector space $X$ to vector space $Y$, $M(0) =0$, and $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$. Does this mean that $M$ is a linear map? If not, could someone please give ...
1
vote
1answer
59 views

Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?

Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation} ...
2
votes
0answers
66 views

Good references on Distribution Theory [duplicate]

Possible Duplicate: Distribution theory book Two books I have been reading are Strichartz's A Guide to Distribution Theory and Fourier Transforms and PartII of Rudin's Functional Analysis . ...
7
votes
2answers
239 views

Is the Laplacian surjective on $C_0^{\infty}$?

Let $M := C_0^{\infty}(\mathbb{R}^n)$ denote the smooth maps with compact support. Then we have a map $\Delta:M\rightarrow M,\,\, f\mapsto \Delta f$, where $\Delta f = \sum_{i=1}^{n} ...
1
vote
1answer
209 views

Discrete Sobolev Space and Sobolev Spaces of Banach Space valued functions

This is a reference request. Can someone kindly give me some refernce(Books/papers) on Discrete Sobolev Space (like we use Discrete $L^p$ spaces of $g\colon\Omega\to\Bbb R $ maps with norm given as ...
1
vote
1answer
109 views

Fatou's Lemma with $\max$

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$ so that $m(W)=1$. For all $k \in \mathbb{Z}_{\geq 0}$ let $f_k: X \times W \rightarrow \mathbb{R}_{\geq 0}$ be measurable and locally ...
1
vote
1answer
40 views

Question about Minimization

Let be $J$ a convex functional defined in Hilbert space H and with real values. What hypothesis I should assume to exist solution for the problem?: $J(u) = \inf \left\{{J(v); v \in K}\right\} , u \in ...
2
votes
1answer
104 views

Decomposition of representations

Let $A$ be a (possibly nonunital) Banach *-algebra, and $H$ be a Hilbert space. If $\pi: A \to B(H)$ is a *-homomorphism, i.e. a representation, then why must $\pi$ be equivalent to a direct sum of ...
1
vote
1answer
191 views

Weak-* convergence and trace-class operators

I am reading a proof by Barry Simon, and he makes a statement equivalent to the following: Let $X$ be the space of compact operators over a Hilbert space $H$. Let $\{y_n\}_n$, $y_n>0$, be a ...
5
votes
1answer
136 views

One problem about complemented subspace

Question: For every Banach space $X$ and its subspace $Y$, is there a complemented subspace $Z$ in $X$ such that $Y \subset Z \subset X $ and $\operatorname{card}(Y)=\operatorname{card}(Z)$ i.e., $Y$ ...
14
votes
2answers
638 views

The spectrum of an unbounded operator

It's well known that the spectrum of a bounded operator on a Banach space is a closed bounded set (and non-empty)on the complex plane. And it's also not hard to find unbounded operators which their ...
2
votes
0answers
75 views

ODE with irregular coefficient

Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution $h(x)$ to be (at least) continuous with its first and second order derivative exist only ...
-4
votes
3answers
261 views

continuity of metric space maps

1.Let $(X_i, d_i)$, $i = 1, 2, 3$, be the metric spaces where $X_1 = X_2 = X_3 =C[0, 1]$ and \begin{align} d_1(f, g) & = \sup_{t \in [0,1]} |f(t) − g(t)| \\ d_2(f, g) & =\int_0^1|f(x)-g(x)|~dx ...
10
votes
1answer
536 views

A Hamel basis for $l^{\,p}$?

I am looking for an explicit example for a Hamel basis for $l^{\,p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one ...
4
votes
1answer
212 views

$L_p$ norm not subadditive for $0<p<1$ when endowed on $C[0,1]$

According to Wikipedia, the $L_p$-norm is not subadditive when $p\in(0,1)$. How can I show that the map $n_p(f)=(\int_0^1|f(x)|^p~\mathrm{d}x)^{2p}$ is not subadditive for $f\in C[0,1]$ for ...
1
vote
1answer
97 views

Relation among $L^{p}(\mathbb{R}^d)$?

Let $L^{p}(\mathbb{R}^d)$ be the linear space consists of $L^p$-integrable functions on $\mathbb{R}^d$ for $1\le p \le \infty$. Are there any relation among these spaces?
3
votes
1answer
213 views

how to show that $c_0$ is complete

I want to show that the metric space $(c_0,d_\infty)$ is complete, where $c_0$ is the collection of all sequences $x\colon \mathbb N\to\mathbb R$ which tend to $0$. I have already shown that the space ...
1
vote
0answers
128 views

Schematic diagram of comparison between mathematical structures

I am looking for diagrams which relate/compare various fields in mathematics or mathematical structures/spaces. My objective is to find out which area of mathematics I should study in order to answer ...
9
votes
1answer
83 views

Self adjoints and unitaries in a banach * algebra

Are the spectra of self adjoints and unitaries in banach * algebras necessarily a subset of the reals and the unit circle respectively? The proofs I know for C* algebras use the continuous functional ...
1
vote
1answer
180 views

spectrum of convex combination

$A,B$ are $n\times n$ Hermitian matrices. If the eigenvalues of $A$ and $B$ are all in an interval $I$, then the eigenvalues of any convex combination of $A,B$ are also in $I$. In the book Bhatia, ...
1
vote
0answers
210 views

equivalence of norms in an open set

I would like to prove that two given norms in the space of smooth functions are equivalent in an open set, is it enough to show that they are equivalent for any compactly contained open set? why? ...
1
vote
0answers
224 views

euler lagrange and gradient descent in level set

the problem began when i read the paper "Active Contours Without Edges" http://www.math.ucla.edu/~lvese/PAPERS/IEEEIP2001.pdf my trouble in equation(9) on page5 derived from page4. At that time,i ...
2
votes
1answer
234 views

inequality between norms of vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. It is known that $\|a\|_{\infty}\leq \|a\|_2\leq \sqrt n\|a\|_{\infty}$. Let $k<<n$. For which kind of vectors the following would be true: $$ ...
3
votes
1answer
148 views

Compact set in all $L_p$, $1\leq p<\infty$

Suppose $X\subseteq L_\infty$ is a compact subset of $L_p$ for all $1\leq p<\infty$. Does this mean that for every $\epsilon>0$ there exists a measurable set $E\subseteq [0,1]$ with ...
1
vote
1answer
63 views

Closure of Locally-Bounded function is Outer SemiContinuous?

Consider a locally-bounded set-valued mapping $f: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ and the set-valued mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ defined as $$ F(x) := ...
1
vote
2answers
79 views

Condition for linear equations to be a contraction

Let $C$ be a fixed $n\times n$ matrix of real numbers and $b \in \mathbb{R}^n$ a fixed vector. Define $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by $y = Tx = Cx+b$. I need to show that, using the ...
0
votes
0answers
93 views

A problem on separable space [duplicate]

Possible Duplicate: Does there exist a linear independent and dense subset? Please bring a hint for following problem: Every separable infinite-dimensional normed space has a linearly ...
1
vote
0answers
82 views

Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ ad every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
0
votes
1answer
238 views

Outer Semicontinuous Mapping?

Consider a locally-bounded function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and a continuous function $g: \mathbb{R}^n \rightarrow \mathbb{R}_{> 0}$. Define the set-valued mapping $F: ...
3
votes
0answers
107 views

Properties of Eigenfunctions of a Kernel

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. I've and Kernel function $K(x,y)$ ...
5
votes
2answers
199 views

Help for Divergence operator

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple. Can some one tell me some reference to study about the invertibility of Divergence operator ...
4
votes
1answer
83 views

Invertible operator not preserving Hilbert dimension

It is known that for a bijective linear operator $T:X\to Y$ the algebraic dimensions of the linear spaces $X$ and $Y$ coincide. I am asking for an example of an invertible (bounded) linear operator ...
1
vote
1answer
504 views

Show that this linear operator is surjective

Let $E$ be a Banach space, and $T$ is a linear operator on $E$, furthermore,it's assumed that $$\sup_{||x||=1}|f(T(x))|<\infty,\forall f\in E^*;$$ $$\inf_{||x||=1}\sup_{||f||=1}|f(T(x)|>0;$$ and ...
1
vote
0answers
445 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
1
vote
1answer
92 views

Does $P^n f(x)=\frac{1}{2^n}\sum_{k=0}^n {n\choose k} f\left(\frac{x}{2^k}\right)$ converge uniformly on compact sets, for $f\in C_b (\mathbb{R})$?

One can show that the Iterated Function System consisting of transformations $$S_1(x)=x, \;\;\ S_2(x)=\frac{1}{2}\;\;\; (x\in\mathbb{R})$$ with constant probabilities $$p_1=p_2=\frac{1}{2}$$ is ...
1
vote
2answers
38 views

How do I solve a function with x^2 and x^-1 to x?

We got two functions: $f(x)=ax^2+b$ $g(x)=x^{-1}=1/x$ I know that they are touching each other in $x=1$. Now I can find out the values for $a$ and $b$ in $f(x)$. Set the derivative of both ...
8
votes
2answers
371 views

Locally integrable functions

Formulation: Let $v\in L^1_\text{loc}(\mathbb{R}^3)$ and $f \in H^1(\mathbb{R}^3)$ such that \begin{equation} \int f^2 v_+ = \int f^2 v_- = +\infty. \end{equation} Here, $v_- = \max(0,-f)$, $v_+ = ...
2
votes
0answers
294 views

A question related to Stone-Weierstrass theorem

Below equation is satisfied. $$ \int_{0}^{\infty} x^nf(x)dx=0 $$ If $n$ is integer with $n\geq0$, then we can't guarantee $f(x) = 0$ for all positive $x$. When $n$ is rational number with $n\geq0$ ...