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, ...

learn more… | top users | synonyms (1)

0
votes
1answer
19 views

Bounded linear space (elementary question)

Does exist (nonzero) bounded normed space over any field? Fix normed linear space $L$ over field K. We have $x+x+x+\cdots\in L$ So $||nx||=n||x||\rightarrow \infty$ when $n \rightarrow \infty \\$ ...
2
votes
0answers
81 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
0
votes
0answers
9 views

infinite compostion of functions: are there generic conditions when this can be done?

I came recently to think about the following problem. Imagine that the followig iteration law is given $$ x_k = f(x_{k-1},u_{k}) $$ If we iterate twice $$ x_k = f(f(x_{k-2},u_{k-1}),u_k) $$ or ...
2
votes
1answer
48 views

What is a integral of operators?

I am reading the book Semigroups of Linear Operatos and Applications to Partial Differential Equations which studies a uniformly continuous semigroup, this is a family $(T_t)_{t \geq 0}$ of bounded ...
0
votes
1answer
44 views

Upper semi-continuity proof for topological spaces

Hi does anyone have any idea or a possible hint for a proof of the following result: Consider asymmetric norm $p$ on $\mathbb{R}$ given by $p(t) = t^{+}$, for $t \in \mathbb{R}$. Show that if $(X, ...
0
votes
1answer
51 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
1
vote
2answers
36 views

How do I prove that there doesn't exist a unit norm vector at a unit distance from a closed subspace of an infinite dimensional vector space?

Let $M$ be a proper closed linear sub space of a normed linear space $X$. If $X$ is finite dimensional, it's a well known result by F.Riesz that there exists a unit vector $x$ such that ...
2
votes
1answer
59 views

Anywhere I integrate $f_n$, the integral approaches $f$. Is $\lim_n f_n = f$ a.e.?

Something tells me this is obvious... I have a bunch of functions: $f,f_n:\mathbb{R}^2\rightarrow \mathbb{R}$, all integrable. Also, $f$ is continuous. I also have a family of sets, $\mathcal{G}$ ...
3
votes
1answer
209 views

Positive-definite function and Positive-definite matrix

I am trying to understand Positive-definite function and read the wikipedia link: https://en.wikipedia.org/wiki/Positive-definite_function It has a relation to Positive-definite matrix and I did not ...
1
vote
0answers
43 views

Centralizer of $C^*$ algebra

Let $\phi: A \to B$ be a surjective $∗$-homomorphism of a separable $C^*$algebra. If $L: A \to A$ is a left centralizer then the formula $\phi(L)(\phi(a)) = \phi(L(a))$ defines a left centralizer for ...
1
vote
1answer
132 views

What would be the “action” in functional analysis?

I am reading Simmons' "Topology and Modern Analysis". He keeps bringing up the idea of studying the set of all structure preserving mappings to obtain info regarding the structure of a certain normed ...
1
vote
1answer
134 views

Is the space of almost everywhere differentiable function with bounded derivative embedded with uniform norm complete?

Let $A$ be the space of almost everywhere differentiable functions $[0,1]\rightarrow [0,1]$, and when differentiable, their derivatives are bounded by $M$. I'm aware that the space of almost ...
0
votes
1answer
29 views

Representation of a Functional Equation.

Let, $f:\mathbb{R}^{J+1}\to\mathbb{R}$, $g:\mathbb{R}^J\to\mathbb{R}$ and consider the following equation: $$f(x,g(x))=g(x).$$ I would like to know if the following statement is true (or find ...
3
votes
0answers
62 views

Hahn-Banach from “Every vector space has basis” [closed]

What is the simplest way to prove Hahn-Banach starting from the AC-equivalent that every vector space admits a basis?
2
votes
0answers
55 views

Span of Polynomials in $\mathcal{C}(\mathbb{R})$ [duplicate]

Let $\mathcal{P}=\{1, x, x^2, x^3 \ldots\}$. Then pick out the correct statements. A) Span$\mathcal{P}=\mathcal{C}(\mathbb{R})$ B) Span$\mathcal{P}$ is a subspace of $\mathcal{C}(\mathbb{R})$ C) ...
0
votes
0answers
54 views

Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps. In the decomposition $$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...
1
vote
0answers
36 views

Help in understanding Bochner's theorem and Pontryagin duality theorem

I am trying to understand Bochner's theorem through wikipedia link https://en.wikipedia.org/wiki/Bochner's_theorem This refers to dual spaces of locally compact abelian group and leads to ...
4
votes
1answer
67 views

Computing the norm $\|f\|$ of a functional.

Define $f: \ell^2(\mathbb{N}) \to \Bbb C$ by: $$f(x) = \sum_{n \geq 1}\frac{x_n}{n^2},$$where $x = (x_n)_{n \geq 1} \in \ell^2(\mathbb{N})$. It is pretty clear that $f$ is linear. Also, since ...
0
votes
1answer
93 views

Hilbert space isometric to a subspace of its dual

Let $\cal H$ be a Hilbert space, and let $\cal H^\ast$ be its dual (of the continuous functionals). If $\cal H$ is a real vector space, I can define: $$\begin{align}\Phi\colon\, &{\cal H} \to ...
1
vote
1answer
29 views

Spectral Measures: Adjoint

This thread is only Q&A! (See the hint: SE: Q&A) Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the ...
2
votes
1answer
35 views

Solving for the spectrum and eigenvectors of the “shift operator(?)” $T$ in $P_3(\mathbb{R})$?

This question is inspired from accidentally misreading this question in my tutorial exercise in my linear algebra course because I forgot the 3 in $P_3(\mathbb{R})$ (The actual question can be ...
0
votes
1answer
15 views

Spectral Measures: Normality

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
3
votes
1answer
80 views

Reference about Sobolev spaces

I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include: general construction ...
1
vote
0answers
45 views

Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
3
votes
1answer
47 views

Do the point-open and compact-open topologies coincide on $C([0,1], \mathbb{R})$?

Do the point-open and compact-open topologies coincide on the space of continuous functions from $[0, 1]$ to $\mathbb{R}$, i.e. on $C([0, 1], \mathbb{R})$? If not, what would be a clear and simple ...
0
votes
1answer
63 views

Why $\sum\limits_{n=1}^\infty \frac{e^{inx}}{n}=-ln(1-e^{ix})$ in $D'$

Functions in D are finite test functions in $C^\infty(\mathbb{R})$ D' are distributions (genralized functions) Do I have to check that $\forall \phi \in D$: $\lim_{\epsilon \to 0} ...
1
vote
0answers
90 views

Total set in a Hilbert space

Definition: A subset of a Hilbert space is total if its span is the entire space. Halmos in his book (A Hilbert space problem book) asks below question: There exists a total set in a Hilbert ...
1
vote
3answers
62 views

Two normal operator that commutes

Suppose $N\in B(H)$ is normal, and $T\in B(H)$ is invertible. Prove that if $TNT^{-1}$ is normal then $N$ commutes with $T^*T$. I can not any idea to prove it, just I know ...
1
vote
1answer
22 views

$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Let $a=\{a_i\}$ be an arbitrary sequence of complex numbers with finitely many non-zero terms. Consider the function $f(x)=\sum_ia_i g(x-i)$, where $g$ is a good function. Prove that for any $p\in ...
0
votes
0answers
21 views

continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
0
votes
2answers
45 views

Minimum curve for the distance between two points at the plane

The problem is to determine the curve y=y(x) in the plane, the lenght of which is given by the functional: \begin{equation} I(y)=\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx=\int_{x_1}^{x_2}F(x,y,y')dx ...
1
vote
0answers
46 views

Help in Understanding the Proof of Baire-Category theorem

In the proof of the Baire category theorem(for non-empty Banach Spaces), I cannot understand the following Baire Category Theorem: A non-empty Banach Space cannot be a countable union of nowhere ...
-1
votes
1answer
58 views

Functionnal analysis: Why $\langle AAx,x\rangle\underset{(*)}{\leq} (\|A\|+m)\langle Ax,x\rangle-\|A\|m ?$

Let $(X,\langle\cdot ,\cdot \rangle)$ an inner vector space and $A\in \mathcal L(X)$ symetric such that $A\geq 0$. I set $m=\inf\{\langle Ax,x\rangle \mid x\in X, \|x\|=1\}$ and thus $A-mI\geq 0$. ...
1
vote
2answers
51 views

Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
2
votes
0answers
46 views

Dimension of a Hilbert space

Halmos in his book (A Hilbert space problem book) says, 1- linear basis, and orthogonal basis of a Hilbert space $H$ have the same cardinality. 2- Also he proves if orthogonal dimension of ...
4
votes
1answer
50 views

Adjoint of $L^{1}$ space

I have a question about $L^{p}$ spaces. Question: Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Let us consider $f \in L^{1}(X)$ satisfying the following property: \begin{align*} \forall ...
0
votes
0answers
50 views

Continuous function on the Skorohod space

I have a process $(X,Y)\in D([0,T],\mathbb{R}^2)$ where $D([0,T],\mathbb{R}^2)$ is the set of cadlag functions with Skorohod metric. Let $A=\{\sup_{t\in[0,T]}|X(t)-\int_0^tX(s)Y(s)ds|>\epsilon\}$ ...
1
vote
0answers
55 views

Frechet derivative of an operator

Let an operator $T:C[a,b]\to C[a,b]$ be defined as: \begin{equation} (Tu)(x)=\int_{a}^{b}K(x,t)f(t,u(t))dt \end{equation} where $K:[a,b]\times[a,b]\to \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to ...
-1
votes
2answers
63 views

Two-sided closed ideals of $C(X,M_2(\mathbb C))$

Let $X$ be compact and Hausdorff space. I know all closed ideals of $C(X)$. I want to substitute $\mathbb C$ by $M_2(\mathbb C)$. What can we say about two-sided closed ideals of $C(X,M_2(\mathbb ...
3
votes
2answers
74 views

Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
1
vote
1answer
26 views

Spectral Measures: Boundedness

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
1
vote
1answer
9 views

unitization of a Banach algebra

For any algebra $A$, the linear space $A_{1}=A+C=\{(a, k)|a \in A, k \in C\}$ equipped with the multiplication $(a,k)(b, l) = (ab+kb+la, kl)$, so-called the unitization of $A$, is a unital algebra ...
2
votes
1answer
42 views

A question in Hahn-Banach theorem

Let $X$ is a real vector space(without topology). call a point $x \in A \subset X$ an internal point of $A$ if $A-x$ is an absorbing set.Suppose $A$ and $B$ are disjoint convex set in $X$ and $A$ has ...
1
vote
0answers
39 views

countable dense set of space of continuous functions on a campact set

Let $X$ be a compact metric space. Let $C_+(X)$ be the set of all non negative continuous functions on $X$. Do there exist a countable dense set of $C_+(X)$? I think the answer is affirmative. For ...
0
votes
0answers
10 views

Compactness in the weak topology generated by dual pair

Let $(X, Y)$ be a dual pair of normed spaces and $\sigma(X, Y), \sigma(Y, X)$ be weak topologies on $X, Y$ respectively. I would like to ask what are the conditions to guarantee $B_X$ compact with ...
0
votes
1answer
77 views

Spectral Measures: Existence

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$Z=N\sqrt{(1+N^*N)^{-1}}\quad ...
0
votes
1answer
35 views

Spectral Measures: Invertibility

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: ...
0
votes
1answer
42 views

Gateaux derivative of a functional $f:\mathbb{R}^2 \to \mathbb{R}$ and why the Frechet derivative of it does not exist

I am more interested in the method which I use for it to be done. I need to understand all the in-between steps so that I can apply it to other examples too. Given the functional: \begin{equation} ...
0
votes
1answer
39 views

Question about norms $p$ and $q$.

I have a simply question: Show that if $x,y \in \mathbb{R^n}$, then $$\biggr|\sum{x_jy_j}\biggr|\leq \| x \|_p \| y \|_q$$ First, I proved that, if $s,t\geq 0$, then ...
1
vote
1answer
62 views

Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...