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)

-2
votes
0answers
31 views

Suppose : $E:R \longrightarrow R$ [closed]

Suppose : $E:R \longrightarrow R$ if $E$ : $$E(a+b)=E(a)E(b): \forall a,b\in R$$ then prove : $$E(r)= E(1)^{r}; \forall r\in Q $$ $Q$= Rational numbers thank you !
1
vote
2answers
53 views

Alternative proof of Fundamental Lemma of Variational Calculus?

I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
3
votes
1answer
62 views

Why are these distributions positive?

I am trying to understand some calculations in a paper by Sidney Coleman. He is showing that certain distributions are positive. The paper can be found here. What I am talking about is happening at ...
1
vote
1answer
37 views

How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
1
vote
0answers
29 views

Structured singular value bound on norm of an inverse operator

My question is related to the first answer here, Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$? with the difference being that I want a similar bound (if possible) with the structured ...
1
vote
1answer
53 views

Bounded operator on Hilbert space

Let $H$ is a Hilbert space. If $T\in B(H)$ show that $T+T^*\ge 0$ iff $T+I$ is invertible in $B(H)$ with $\|(T-I)(T+I)^{-1}\|\le 1$. (Hint is $T+T^*\ge 0$ iff $\|(T+I)x\|\ge \|x\|\ $ and ...
1
vote
0answers
36 views

Finding the closest function to another in a Hilbert space.

Let H be the Hilbert space L$^2$([0,1)], and let $S$ be the subspace of functions f $\in$ H satisfying $\int^1_0(1+x)f(x)dx=0$. Find the element of $S$ closest to the function $g\in H$ defined ...
1
vote
2answers
36 views

What is Convex about Locally Convex Spaces?

This might be a silly question, but what motivates the name "locally convex" for locally convex spaces? The definition in terms of semi-norms seems to have nothing to do with convexity or with the ...
-2
votes
0answers
35 views

Can I have a map to follow in this proof?

Let X be a normed vector space, $Y:=B(X,\mathbb{K})$ a vector space over $\mathbb{K}$ of the bounded linear functionals with the usual norm $$||f||=sup_{||x||=1, x\in X}|f(x)|, f\in Y$$ and$$F(x) ...
0
votes
1answer
49 views

Cardinality of a Hilbert space

I have seen the theorem about the cardinality of orthonormal basis of a Hilbert space. I wonder if we have a Hilbert space $H$ with an orthonormal basis having cardinality of the continuum, then what ...
1
vote
0answers
26 views

Schmidt decomposition problem

I'm having a problem in implementing the following problem: I have a quantum state so defined: $\left| \Psi\right>=\int ...
2
votes
1answer
23 views

Convexity of Hilbert cube [on hold]

I am trying to show that the Hilbert cube $\{ x_n \in l^2(\mathbb{N}) \mid x_n \in [0, \frac{1}{n}] \ \forall n \in \mathbb{N} \}$ is convex and (norm)-compact.
1
vote
2answers
427 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
2
votes
2answers
39 views

If for $u \in L^2(\mathbb{R}^n)$ , we define $v(t)=u(x+th) $ $v: [0,1] \to \mathbb{R}$ $\Rightarrow^?$ $v \in L^2((0,1))$

I have a function $u(x) \in L^2(\mathbb{R}^n)$ ($n \geq 2$). Suppose we define another function $v$ as $$v:[0,1] \to \mathbb{R} $$ $$\quad \quad \quad \quad \quad \quad \ t \to u(x+th)$$ where $h \in ...
3
votes
2answers
602 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
5
votes
1answer
93 views

Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
5
votes
1answer
76 views

Topological vector spaces book recommendation

I'm currently taking a class covering the theory of topological vector spaces using the book Topological Vector Spaces, Distributions, and Kernels by Francois Treves. I find the subject to be very ...
1
vote
1answer
19 views

Closed Graph Theorem on Finite Dimensional Banach Spaces

My professor wants us to prove that every linear mapping from a finite dimensional Banach space is continuous. BUT, he wants us to do so using the Closed Graph Theorem (in functional analysis). ...
0
votes
1answer
34 views

How to evaluate the Lebesgue integral of the Heaviside function?

I have to evaluate the Lebesgue integral $$ I = \int\limits_{[-1, 1]} \chi(x) \chi(x - \frac{1}{2}) d\left(\chi(x)\chi(x + \frac{1}{2})\right) $$ where $ \chi $ is the Heaviside function: $$ ...
0
votes
1answer
20 views

Closest element to a subset of $\mathbb R^2$

Let $U=\{(x,y)|x,y\geq 0\}$ be a closed convex subspace of $(\mathbb R^2,\|\cdot\|_\infty )$. Show that the closest elements in $U$ to $(1,-1)$ are $\{(x,0)|0\leq x\leq 2\}$ Show that the closest ...
2
votes
1answer
90 views

What does $ f ^ {n} (x ^ {1/n}) = … $ mean?

I was asked to check whether the sequence of functions $ \{ x_{n} (t) \} $ defined as $$ x ^{n} _{n}(t ^ \frac{1}{n}) = \begin{cases}n, & t \leq \frac{1}{n} \\\frac{1}{n},& t > ...
1
vote
0answers
34 views

Convergence on $\ell^p$ spaces

I have been working on this problem. Show the following are equivalent: i. $\sup_{n\in\mathbb{N}}\|x_n\|_p<\infty$ and $\lim_{n\to\infty}x_n(j)=0$ for each $j\geq1$, ii. ...
-1
votes
0answers
15 views

On bounded linear operators not attaining its norm

Suppose $T$ is a bounded linear operator on a Hilbert space $H$ which does not attain its norm. Does there exist a sequence $\{e_n\}_{n\in\Bbb N}$ of orthonormal vectors such that $\lVert Te_n\rVert ...
0
votes
0answers
17 views

Can dimension counting argument generalize to functional space

By dimension counting I mean the following argument: there is no injective continous mapping from any open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ if $n>m$ (is this true? I can't give rigous ...
5
votes
2answers
63 views

Weak convergence in different $L^p$ spaces

Consider $p \ge \alpha \ge 1.$ If a sequence converges weakly in $L^p,$ say $u_n \rightharpoonup u$, is it true that: $$u_n^{\alpha} \rightharpoonup u^{\alpha} \text{ in $L^{p/ \alpha}$}$$ This ...
2
votes
1answer
64 views

Inequality on a general convex normed space

Assume $(X,\|\cdot\|)$ is a normed space with the following property: if $x \neq y \in X$ have norm 1 then $\|\frac{x+y}{2}\|<1$. (We then say that $X$ is strictly convex) Prove that if $C$ is a ...
0
votes
1answer
25 views

Finding the closest element to a function in a normed space containing functions.

Let $B=\{f\in c[1,0]|\forall 0\leq x\leq 1 : f(x) \geq 0\}$. Given $f\in c[0,1]$, find the closest element in $B$ under the $\|\cdot\|_2$ norm. I can see something similar to this question in the ...
3
votes
1answer
145 views

Show that if $[Q,P]=it\Bbb{I}$ then the operators are unbounded

In the Hilbert space $\mathcal{H} = L^2(\mathbb{R},dx)$, let 2 symmetrical operators $P$ and $Q$ be given, with the following properties: $D(P) = D(Q) = \mathcal{S}(\mathbb{R})$ ...
1
vote
0answers
33 views

Reference Quest: Measure Theoretic and Functional Analytic Intro to Stochastic Processes

Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, ...
0
votes
0answers
16 views

Applications of Stein's Interpolation Theorem

Are there any neat applications for Stein's interpolation theorem, especially in the context of complex analysis? There's a lot of proofs online but I couldn't find many "corollaries" that follow from ...
4
votes
1answer
38 views

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$?

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$? I've seen this answer but this is on an infinite domain. I'm interested only in $(0,1)$. I tried playing around with ...
4
votes
1answer
69 views

Is $C^\omega([0,1])$ normable? (And about the growth of coefficients of infinitely differentiable functions)

This question arised to me when trying to prove that the space of infinitely differentiable functions defined in a compact space $K\subset\mathbb{C}$ taking values in $\mathbb{C}$, that is ...
14
votes
1answer
850 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
0
votes
1answer
11 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
0
votes
1answer
30 views

Hahn Banach theorem proof

Let $X$ be real linear space and $X_{0}$ is its subspace. Also let $p$ be a finite convex functional in $X$ and $f_{0}$ is linear functional in $X_{0}$, while $f_{0}(x)\le p(x),x\in X_{0}$ Then ...
0
votes
3answers
41 views

Definition of a closed subset of $\mathbb R$

Let $A=\{x\in \mathbb R \mid 0\leq x\}$. Prove that $C$ is a closed subset of $\mathbb R$. As far as I understand a closed set/subset is a set that for all sequences within it are bounded and the ...
0
votes
0answers
15 views

In separable, inner-product space X every complete orthonormal set is closed and vice versa

English is not my native language so if anything needs definition just tell me. I have a problem with proving that if set is closed then it is complete Let $\{\varphi _{k}\}_{k=1}^{\infty }$ be an ...
3
votes
0answers
47 views

proving compactness with arzela-ascoli

Let $C>0$ and $$ M=\left\{f\in C^1[0,1]:\int_0^1|f(x)|^2dx+\int_0^1|f'(x)|^2dx\leq C \right\} $$ Is $\overline M$ compact in $C[0,1]$? I think this follows by the Arzela-Ascoli theorem. How can ...
0
votes
2answers
27 views

Bessels inequality

Why can you change the top index of sum in the last step of this proof (see below) from $n$ to infinity ? Let $r_n = x - \sum_{k=1}^{n} \langle x,e_k \rangle \cdot e_k$. Then for $j = ...
0
votes
1answer
9 views

Weakly square summable series as operators on Hilbert spaces

Let $H$ be a Hilbert space and let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence in $H$ such that $\sum^{\infty}_{n=1}|\langle h,a_n\rangle|^2<\infty$ for all $h\in H$. Here ...
12
votes
1answer
447 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
4
votes
1answer
57 views

proof of an equality norm

Let the mapping $T:\ell^{2}\rightarrow \ell^{2}$ is defined as follow. $$T(x_1,x_2,\ldots,x_n,\ldots)=(x_1,\dfrac{1}{2}x_2,\ldots,\dfrac{1}{n}x_n,\ldots)$$ In this case, i've easily earned: ...
0
votes
0answers
10 views

Control the value of a function at a point by the norm of its fourier transformation and itself

$n\leq 3$ ,$\Delta$ is the Laplacian on $L^{2}(R^{n})$, $Dom(\Delta) = \{\phi\in L^{2}(R^{n})|\Delta\phi\in L^{2}(R^{n})\}$. Please show that:for any $\phi\in Dom(\Delta)$,there exists constants ...
1
vote
1answer
26 views

Theorem 2.14 (The dual of $L^p(\Omega)$) in Lieb's Analysis book

The following pictures are Theorem 2.14 (The dual of $L^p(\Omega)$ in Lieb's Analysis book and its proof of the case $1<p<\infty$. My question is how to get the inequility (3) in the red box? ...
0
votes
0answers
17 views

What is the discretization matrix of 2D Poisson equation of finite diffence with checkerboard (black and red) pattern?

Given the problem$-\Delta u(x,y)=f(x,y)$ on unit rectangle $\Omega=[0,1]^{2}$ and $u(x,y)=g(x,y)$ on $\partial\Omega$, what is the finite difference matrix associated with step size $h=1/(2N+1)$ where ...
0
votes
0answers
24 views

Proof that addition on a Banach space is continuous

What I have so far: Let $(W,+,\cdot,\Vert\cdot\Vert)$ be a Banach space. We have the map $$+ : W\times W \to W.$$ The topology $T$ on $W$ is given by $$T = \{U\subseteq W ~\big|~ \forall u\in U : ...
0
votes
1answer
24 views

A continuous action of a compact group on a uniform space is equicontinuous?

I am stuck on how to prove the statement "A continuous action of a compact group $G$ on a uniform space $X$ is equicontinuous." So, essentially we want to show that for every entourage $\alpha$ of ...
4
votes
1answer
36 views

Spectrum of linear operator, essential spectral radius

Consider the operator $L:L^1(S^1)\to L^1(S^1)$ given by $$ (Tf)(x)=\dfrac{1}{2}\left( f\left( \dfrac{x}{2}\mod 1\right)+f\left( \dfrac{x+1}{2} \mod 1 \right) \right) $$ where we identified $S^1$ with ...
0
votes
2answers
65 views

Real Analysis , Folland Problem 6.1.5

Problem 6.1.5 - Suppose $0 < p < q < \infty$. Then $L^p \not\subset L^q$ if and only if $X$ contains sets of arbitrary small positive measure, and $L^q\not\subset L^p$ if and only if $X$ ...
2
votes
1answer
21 views

GNS-Construction: Involution

Given a C*-algebra $\mathcal{A}$. (It may or may not contain identity!) Consider a positive linear functional: $$\omega:\mathcal{A}\to\mathbb{C}:\quad A\geq0\implies \omega(A)\geq0$$ Construct its ...