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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8
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1answer
280 views

Short and elegant introduction to Sobolev spaces

I am preparing a course on Nonlinear Analysis, and I need to teach the most important facts about Sobolev spaces to my students. I know most books on this subject, from Brezis' to Adams', from Mazya's ...
2
votes
0answers
66 views

Invertibility of a matrix

I have to study the invertibility of this matrix $$A(k)=\bigg[\bigg(\alpha_j-\frac{ik}{4\pi}\bigg)\delta_{jj'}-\tilde{f}(y_j-y_{j'})\bigg]_{jj'}$$ where $\tilde{f}(x)$ is ...
5
votes
1answer
94 views

Why locally compact in the Gelfand representation?

I'm missing something in the Gelfand representation. Let's just say $\mathfrak{A}$ is a Banach algebra. Then it's a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear ...
3
votes
2answers
997 views

Spectral radius of the Volterra operator

The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: ...
0
votes
1answer
38 views

Computing a derivative of map from $V \to V^*$ (PDEs and regularity)

I am reading Rogers and Renardy book on parabolic regularity. There they consider a PDE $$\dot u = A(t)u + f(t)$$ where $A(t):V \to V^*$ is an operator. In the regularity result, they need $A \in ...
2
votes
1answer
100 views

Calculation of the Fourier transform of a function

I have calculated the Fourier transform of this function $$f(x)=\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}$$ with $x\in\mathbb{R}^3$, $\Im \sqrt{z}>0$ e $y$ fixed point in $\mathbb{R}^3$. I have obtained ...
0
votes
1answer
118 views

Prove that there is a subsequence of functions which converges uniformly

The problem is this Let $\phi:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$ be bounded and continuous, and for $n=1,2,\dots$ let $f_n:[0,1]\rightarrow \mathbb{R}$ satisfy $f_n(0)=1/n$ and ...
1
vote
0answers
52 views

Sum of Hilbert spaces

For various reasons I need to define the following space $$\hat{H}=\{u\mid u=f+g, \;f\in H^{2,-s}(\mathbb{R}^3),\; g\in L^{2,-s}(\mathbb{R}^3)\}$$ where ...
1
vote
1answer
32 views

Question about a proof.

I was just reading over this proof here Adjoint identity . I am just learn functional analysis, and I wonder, in the step where it says "Since $D(A)$ is a subspace we must have $f(v,Av)=0$", why this ...
3
votes
3answers
205 views

distribution with point support

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
9
votes
2answers
2k views

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each ...
10
votes
1answer
498 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
6
votes
1answer
675 views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
1
vote
1answer
63 views

Find a decoupled explicit formula for a minimizer

Consider the energy $F(u,v) = \int^1_0((\frac{1}{4}(u')^2+(v')^2 +\frac{1}{2}(u-v+1)^2)dx$ for $C^1$ functions u and v on the interval (0,1) that satisfy the boundary conditions ...
3
votes
1answer
138 views

Is the unit ball: $B(0,1)=\{f \in L_p(X,u): \|f\|_p<1\}$ convex? , $0<p<1$

Let $(X,\mathbb{X},u)$ be a measure space $L_p(X,u)=\{ f:X\to \mathbb{C}: \|f\|_p<\infty\}$ , $0< p <1$ , $f$: measurable function $\|f\|_p=\left( \displaystyle \int_X |f|^p ...
4
votes
1answer
218 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
3
votes
1answer
189 views

The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
5
votes
0answers
65 views

$e^{iBt}e^{-iAt}$converges as operator norm

Let $A,B$ be self-adjoint operators on $H$,then we can define the strong limit $$ W=s-\lim_{t\to+\infty}e^{iBt}e^{-iAt} $$ If the limit exsists, then W is called the wave operator, which is ...
3
votes
1answer
156 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
2
votes
1answer
145 views

linear transformation between spaces of continuous function on metric space

Let $M_1$ and $M_2$ be compact metric spaces. Denote $C(M_i)$ as the space of continuous functions from $M_i$ to $\Bbb C$ with supremum-norm, $i=1,2$. A linear function $T:C(M_1)\to C(M_2)$ is said ...
6
votes
1answer
97 views

$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$
3
votes
1answer
356 views

Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.

If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then $ a = u|a| $ for a unique unitary element $ u $ of $ A $. If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $? I ...
11
votes
1answer
531 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
4
votes
3answers
403 views

Let $ (x_n) $ be a divergent sequence in a compact subset of $ \mathbb{R}^n $. Prove there are two subsequences that converge to different limits.

Let $(x_n)$ be a divergent sequence in a compact subset of $\mathbb R^n$. Prove that there are two subsequences of $(x_n)$ that are convergent to different limit points. Some ideas that might be ...
1
vote
1answer
167 views

Question on Continuous function and Lipschitz

$f:\mathbb{R}^n \to \mathbb{R}$ is contiuous. If $x \in \mathbb{R}^n$ and $C \in \mathbb{R}$ such that $f(x) < C$ ($C$ is constant). Prove that there is $r>0$ such that $\forall{y} \in B_r ...
2
votes
1answer
123 views

using uniform boundedness principle

I have a sequence of numbers $x_n$ that satisfy that for every $y_n \in c_0$ (when $c_0$ is a Banach space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{a_n} =0$ ) the series ...
1
vote
1answer
84 views

Range of the generator of a one parameter semigroup of operators?

From Wikipedia: If $X$ is a Banach space, a one-parameter semigroup of operators on $X$ is a family of operators indexed on the non-negative real numbers $\{T(t)\} t ∈ [0, ∞)$ such that $$ ...
2
votes
1answer
622 views

Minimizing continuous, convex and coercive functions in non-reflexive Banach spaces

Let $X$ be a infinite dimensional real Banach space. If $X$ is reflexive, then any continuous, convex coervive function $f:X\rightarrow\mathbb{R}$ has a minimum value, that is assumed for some point ...
3
votes
1answer
227 views

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup?

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup? Isn't $C_0$ the set of continuous functions that vanish at infinity? Thanks and regards!
4
votes
2answers
183 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
6
votes
1answer
1k views

Is “Functional Analysis” by “Yosida” a good book for self study?

I was wishing to start studying by myself the book Functional Analysis by Yosida, does anyone have already used it, is it a good reference?
0
votes
1answer
147 views

When is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$? [duplicate]

Given a measure space $(X,\Sigma,\mu)$, when is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$ for $p>r$, or for $p<r$ . Thanks.
0
votes
1answer
77 views

A function not in $L^2(\mathbb{R}^3)$

From this equation $$ (p^2-\alpha)\hat{f}(p)=\frac{e^{-ip\cdot y}}{p^2+\lambda}$$ where $\hat{f}$ is the Fourier transform, $\alpha,\lambda>0$ e $y$ a fixed point in $\mathbb{R}^3$ can I conclude ...
2
votes
1answer
118 views

showing subset of $l^2$ closed

I was looking for an example of a bounded and closed set which is not compact. Considering $l^2$ and looking to a set $K$ of canonical basis $e_i=(0,...,1_i,...,0)$. This is bounded, Is it true that ...
1
vote
1answer
119 views

Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$

Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
0
votes
1answer
93 views

Distance of point for a set in linear spaces

Let $X$ a normed linear spaces, $Y \subset X$ a subspace and $z \in X$ an arbitrary point. How can we show that: $$\text{dist} (z, Y) = \sup \{\psi(z) \ | \ \|\psi\| = 1, \psi \equiv 0 \ \text{on} \ ...
1
vote
1answer
74 views

Equintinuity of bounded linear functions equivalent to uniform boundedness

The claim is the following: Every family of bounded linear functions is equicontinuous if and only it is uniformly bounded. Equicontinuity is defined here. Any suggestions about this? I only ...
0
votes
1answer
106 views

Laplacian in $\mathbb R^3$ of the function $\frac{1}{|x|}$

In an exercise I find the request of evaluating the Laplacian of $f(x)=\frac{1}{|x|}$ in $\mathbb{R}^3$. But it exists in a classical sense only if $x\neq 0$ otherwise I have to see it in ...
8
votes
3answers
565 views

Weak convergence in reflexive Banach space

Let $X$ be a reflexive Banach space. Let $T: X \to Y$ a linear operator. I want to show that: $$T \in \mathcal{L}(X, Y) \iff ((x_n \stackrel{w}{\rightharpoonup} x) \implies (T(x_n) ...
0
votes
1answer
54 views

Homogeneous function of degree $-1$

If I have a bounded and homogeneous function of degree $-1$, can I conclude that it goes to $0$ pointwise at infinity, isn't it?
6
votes
1answer
328 views

On the spectrum of the sum of two commuting elements in a Banach algebra

Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que a*b=b*a. Pourquoi σ (a+b) с σ(a)+σ(b) Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)? Translation: Let ...
1
vote
2answers
91 views

Inequality regarding weak-* convergence

Let $X$ be a normed linear space, $\psi \in X^{*}$ and $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ a sequence in $X^{*}$. Show that if $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ converges weak-${*}$ to ...
3
votes
2answers
84 views

Norm of the operator $Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt$

Consider the operator $T:(C[-1, 1], \|\cdot\|_\infty)\rightarrow \mathbb R$ given by, $$Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt,$$ is $\|T\|=2$. How to show $\|T\|=2$? On the one hand it is easy, ...
1
vote
1answer
231 views

characterization of Euclidean norm by the parallelogram identity

can any one give me a prove to the following: A norm is Euclidean iff satisfies the parallelogram identity, $\|v+w\|=\sqrt{2\|v\|^2+2\|w\|^2-\|v-w\|^2}$
22
votes
3answers
635 views

Is $\mathbb{R}^{\infty}$ homeomorphic to $\mathbb{R}^{\infty}\setminus\{0\}$?

Let $\mathbb{R}^{\infty}$ be a linear topological space of all sequences $x=(x_{1},x_{2},\ldots,x_{n},\ldots)$ of real numbers with a product topology, or, in other words, let $\mathbb{R}^{\infty}$ be ...
1
vote
1answer
294 views

$X \subset C^0[0,1] $,closed, bounded, equicontinuous $\implies $ $\forall g \in \! X \quad \exists h \in X$ s.t.$ \int_0^1 h \geq \int_0^1 g$

I took a Real Analysis class and have been teaching myself some basic functional analysis. I am really unsure of how to prove the following: Let $X \subset C^0[0,1]$ that is closed, bounded, and ...
0
votes
1answer
61 views

Does this problem make sense? “Give an example of a set $F\subset C([0,1])$ which is pointwise bounded but not bounded”

I think that the professor might mean pointwise bounded but not uniformly bounded? Or is there a way that it makes sense to think of $F$ being bounded? $C([0,1])$ refers to the set of continuous ...
5
votes
1answer
343 views

Spectral radius in Banach Algebra

Let $A$ be a unital Banach algebra and $a\in A$ and $\lambda \in \rho(a)$. I want to prove that $$r(R(a,\lambda))=\frac{1}{d(\lambda,\sigma(a))}.$$ where $R(a,\lambda)=(\lambda 1-a)^{-1}$ and $r(.)$ ...
2
votes
1answer
119 views

Self-adjoint operator and inner product

I am wondering whether there is a way to make sense of self adjointness of an operator on $C[0,1]$ without resorting to the inner product of $L^2[0,1]$. I am not referring to concrete alternative ...
6
votes
1answer
312 views

Is the Sobolev embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C_0(\mathbb{R}^d)$ compact?

In p. 508 of the paper : http://www.jstor.org/stable/2243484 , it is mentioned that if $2l \geq d$, the embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C_0(\mathbb{R}^d)$ is compact, where ...