Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

3
votes
0answers
35 views

Adjoint of Bounded Below Operator

Let $H$ be a complex, separable Hilbert space, and $T:H \rightarrow H$ a linear, bounded operator. Assume that $$\sigma(T) = \sigma(T^*) = \{ \lambda \in \mathbb{C}: a \leq |\lambda| \leq b \}$$ for $...
-1
votes
1answer
33 views

How to define $L^{\infty}$ norm for vector-valued functions?

for functions $f: \mathbb{R}^{n} \rightarrow \mathbb{R} $ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$, $\| f-g\|_{\infty} := \sup_{x\in \mathbb{R}^{n}}\|f-g \|$. But if now we have $f: \mathbb{R}^{...
0
votes
0answers
23 views

Different topologies in Sobolev space $W^{1,p}$

In paper [1], L.Ambrosio talks about the space $W^{1,p}(\Omega),\ 1\leq p<+\infty$ endowed with four different topologies: The strong topology, denoted by $W^{1,p}(\Omega)$. The weak topology, ...
2
votes
0answers
33 views

Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
2
votes
0answers
43 views

Two PDE for one matrix-valued unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $P = P(x,t) \in \mathbb{R}^{3\times 3}$, $Q = Q(x,t) \in \mathbb{R}^{3\times 3}$, $R_0 = R_0(x) \in \mathbb{R}^{3\times 3}$ and $G= G(t) \in \mathbb{R}^{3 \...
0
votes
0answers
33 views

Extension of Choi's theorem

In the paper written by Man Duen Choi "Completely Positive Linear Maps on Complex Matrices", there was a criterion mentioned/theorem. For reference, I have written it below. Let $ϕ:M_n→M_m$. Then ϕ ...
1
vote
0answers
18 views

Weak star convergence of Borel probability measures on a metric space

Let $(X,\rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $\sigma$-algebra of $X$. Suppose $\mu_n,\mu \in P(X)$ for $n \in \mathbb{N}$ such that $\...
0
votes
0answers
32 views

The span of a finite number of vectors in a normed vector space is closed

I want to prove that the span $S$ of a finite number of vectors $v$ in an arbitrary normed vector space $V$ is a closed set. My plan is to show that all convergent sequences ${x_n}$ contained in the ...
0
votes
0answers
18 views

Simple Consequences of Goldstine's Theorem

For a normed space $ X $, let $ J : X \to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $, and let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ ...
0
votes
1answer
25 views

Absolutely continuous Banach space valued function

Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the ...
0
votes
0answers
30 views

Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$ \|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
1
vote
1answer
36 views

If operator is closed and densely defined then $D(A^*)^\perp = \{0\}$

I'm a bit rusty in my Functional Analysis and couldn't solve this question: Let $X$ be a Banach space (over either $\mathbb{R}$ or $\mathbb{C}$) and $X^*$ its dual space. Show that, if $A:D(A) \...
0
votes
0answers
32 views

Why is $ \max_{i} | \lambda_i(A) | \leq \| A \|_P $?

I was told: $$ \max_{i} | \lambda(A) | \leq \| A \|_P $$ I tried thinking through it. So the operator norm is defined as: $$ \| A \|_P = \sup_{y \neq 0} \frac{ \| A y \|_P }{ \| y\|_P } = \sup_{ \| ...
0
votes
0answers
30 views

Rewriting a $\max\left\{0,\dots\right\}$ function in order to integrate the function more properly

Yesterday I asked a question about a certain integral. In the integral is the term: $$\max\left\{0,\left|\text{n}\cdot\sin\left(2\pi\cdot x\cdot t-\frac{\pi}{2}\right)\right|-2\cdot\text{z}\right\}\...
0
votes
0answers
41 views

Couldn't Derive the Result Given in a Book

Currently I'm trying to read Stroock's "lectures on topics in stochastic differential equations" book. In lemma $2.5$, he states that given $f$ and $g$ that are strictly increasing function on $[0, \...
4
votes
2answers
61 views

Uniqueness in $C([0,T])$ of solution found by Picard-Lindelof Theorem

Full statement of the question: Suppose $\alpha \in \mathbb{R}$, also $f: \mathbb{R} \rightarrow \mathbb{R}, f \in C^{1}(\mathbb{R}), and f(0) = 0$ Consider the following ODE: $\partial_{t} u(t) =...
0
votes
1answer
41 views

Difficulty Understanding Sufficient Conditions for Weak Extrema in Calculus of Variations

I am having a difficult time understanding Jacobi's necessary condition for weak extrema of functionals. Graphics and detailed explanations would be helpful. I am following the following two texts: ...
2
votes
1answer
39 views

If $ f*g$ is a polynomial of degree at most $m$ for all $g \in C_{0}^{\infty} $. Show $f$ is a polynomial of degree at most $m$ almost everywhere

$C_{0}^{\infty}$ denotes the set of smooth functions with compact support. In attempt of this, I've evaluated the convolution at the (m+1)th derivative to obtain $$ \frac{d}{d^{m+1}}(f*g) =\int f(x-y)...
0
votes
0answers
24 views

Weak convergence and compacity

Please I dont understand this. I have: $ \parallel \nabla m_n \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C$ $ \parallel \frac{\partial m_n}{\partial z} \parallel_{L^{\infty}(\mathbb{R}^+,...
0
votes
1answer
43 views

Uniform inequality for a continuous function

Let $f(x,y)\in \mathcal{C}([a,b]\times[c,d])$ such that $$\exists \xi\in (a,b) : f(\xi,y)\neq 0, \forall y\in [c,d].$$ By the continuity of $f$, we have $$|f(\xi,\cdot)|\geq \min\limits_{[c,d]} |f(\...
2
votes
0answers
33 views

What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
2
votes
0answers
34 views

For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement: For a compact Riemannian manifold $M$ and the corresponding Laplace-Beltrami operator $\Delta$ on $M$ we have, that $$L^2(M) = \widehat{\bigoplus_{\...
2
votes
0answers
18 views

Shift operator on the double-size Hilbert space $\ell^2(\mathbb{N}^*)\oplus \ell^2(\mathbb{N}^*)$

It is well known that, the right shift operator is given by \begin{align*} A_1\colon \ell^2(\mathbb{N}^*) & \rightarrow \ell^2(\mathbb{N}^*) \\ (x_1,x_2,\cdots)&\mapsto (0,x_1,x_2,\cdots), \...
1
vote
1answer
16 views

Show that resolvant is analytic outside the spectrum

Let $T$ be a bounded operator on $Hilbert$ space $\mathcal{H}$. Show that $R_{\lambda}=(T-\lambda)^{-1}$ is an analytic function on open set $\rho(T)=\mathbb{C}\setminus sp(T)$. I know $R_{\lambda}=-\...
2
votes
1answer
41 views

The tensor product of two blocks of positive operators is positive

Let $$T = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \end{bmatrix},\quad S = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix}$$ be two positive operators on $E\...
0
votes
1answer
52 views

Prove $\|T\| = \sup_{\|x\| < 1} \|Tx\|$

Let $X, Y$ be Banach spaces. And $T \in B(X\rightarrow Y)$. Prove that $$\|T\| = \sup_{\|x\| < 1} \|Tx\|$$ Discussion Having trouble seeing how to handle some of these ideas below. Please let me ...
1
vote
0answers
24 views

Weak Convergence from Strong Convergence

Let $\Omega \subset \mathbb{R}$ be a bounded domain, $v \in H_{0}^{1}(\Omega)$ such that $||u_{n}-v||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$ for a bounded sequence $\{u_{n}\}_{n\in\mathbb{N}} \...
3
votes
2answers
40 views

Deforming a linear map a little preserves surjectivity

Let $X$ be a Banach space, and $A: X\to X$ be a surjective linear map. Define $$\eta(A) = \{\lambda\in \mathbb{C}: A - \lambda I \text{ is surjective}\}$$ where $I: X\to X$ is the identity. Show that $...
0
votes
2answers
49 views

What is the Geometric meaning of vector norm in Rn n>3

My question is related to the length of the vector , Sorry it may seem stupid for you as i come from engineering background not mathematics background For Vectors up to 3 dimensions (can be ...
1
vote
1answer
36 views

An identity for Fourier transform of measure

Consider a finite Borel measure $\mu$ on $\mathbb R$. The Fourier transform $\hat{\mu}$ of $\mu$ is defined by $\hat{\mu} (\xi)= \int _{\mathbb R} e^{-ix\xi} d\mu(x)$. I would like to prove the ...
0
votes
2answers
36 views

Explaining the Proof of Schwarz Inequality for Scalar Product in a Vector Space

Let $\langle x,y\rangle$ be the scalar product of $x$ and $y$ in a linear space $X$ over either $\mathbb{R}$ or $\mathbb{C}$. This scalar product satisfies the three properties: Bilinerity/...
6
votes
2answers
285 views

Non-Borel set in arbitrary metric space

Most sources give non-Borel set in Euclidean space. I wonder if there is a way to construct such sets in arbitrary metric space. In particular, is there a non-borel set in $C[0,1]$ all continuous ...
1
vote
0answers
47 views

Invariant subspace for Volterra operator

So, am I stupid or isn't this kind of trivial? I'm having this problem: Let $T$ be the Volterra operator on $L^2([0,1])$ defined by \begin{equation} Tf(s) = \int^s_0 \, f(t) \, \text{d}t. \end{...
1
vote
1answer
42 views

$L^p$ functions for $p$ in $[a,b]$

I have seen a theorem that assures that the set $A=\lbrace p\in [1,\infty]: u\in L^p(0,\infty)\rbrace$ is an interval. It is easy to find a function $u$ for which $A$ is the empty-set or $[1,\infty]$ ...
0
votes
0answers
36 views

Show that the operator associated to a spectral decomposition on a Hilbert space is self-adjoint

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
2
votes
0answers
36 views

Question: can we say that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$?

Let $\Omega$ be a compact manifold in $\mathbb R^2$. For $1 \leq p \lt 4/3$ can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$ with the first inclusion being ...
0
votes
2answers
23 views

Convergence for the norm of a sequence in $\ell^{\infty}$

I still don't know if the statement below is true. It seems to be true but I couldn't find a proof. The statement is the following: for each $n \geq 1$, let $x_n = (a_1(n),a_2(n),\ldots,a_n(n),0,0,\...
2
votes
1answer
26 views

$G$ has Kazhdan's property (T) $\iff$ $G$ has a Kazhdan pair

A locally compact group $G$ is said to be Kazhdan or have Property (T) if for any unitary representation $\rho$ that has almost invariant vectors (a.i.v) it has an invariant vector. Meaning of a.i.v -...
0
votes
1answer
26 views

Krein - Smulian theorem - norm or weak closure?

The theorem of Krein - Smulian reads as follows: The closed convex hull of a weakly compact subset of a Banach space is weakly compact. We consider the closed convex hull - but is it norm closure or ...
2
votes
1answer
54 views

Are $(\ell^\infty(\mathbb{Z}))^*\simeq (\ell^\infty(\mathbb{N}))^*$ isomorphs?

Are $(\ell^\infty(\mathbb{Z}))^*$ and $ (\ell^\infty(\mathbb{N}))^*$ isomorphs? I think that I could establish the next function $\Phi:(\ell^\infty(\mathbb{N}))^*\to(\ell^\infty(\mathbb{Z}))^*$ such ...
3
votes
0answers
35 views

Existence of functionals on $L^0$

Studying a paper about risk measures by F. Delbaen, I bumped into this statement: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space: if $\mathbb{P}$ is atomless, then there exists no ...
1
vote
0answers
18 views

Is the approximate point spectrum simply the union of the essential and point spectra?

I'm refreshing functional analysis from "Hilbert space operators in quantum physics" by Blank, Exner and Havlíček (Springer, 2008). They define essential spectrum for any closed operator on a Hilbert ...
2
votes
0answers
45 views

Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...
3
votes
1answer
48 views

Are $c_0$ and $c$ duals of some spaces?

The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true. That is, not all Banach spaces are isomorphic to the dual space of some normed vector space. ...
2
votes
1answer
15 views

Completeness of a Normed Space of Smooth, Bounded Functions

As part of a proof of the Picard–Lindelöf theorem, I am using the following space: $X = \{ u \in C([0,T]) : u(0) = \alpha , || u - \alpha || \leq K\}$ where $K \in \mathbb{R}_{> 0} , \ \alpha \in ...
1
vote
1answer
26 views

Show that the generator of a strongly continuous contraction semigroup on $L^2$ is nonpositive definite

Let $(E,\mathcal E,\mu)$ be a finite measure space, $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $L^2(\mu)$ and $(\mathcal D(A,A)$ denote the generator of $(T(t))_{t\ge0}$. ...
1
vote
3answers
77 views

Why is operator norm defined the way it is?

Is there an intuition for the infimum definition of ${\| A \|}_{\mathrm{op}}$ without using a different, equivalent definition? I am referring to the definition, given an operator $A: W \rightarrow V$...
4
votes
1answer
37 views

Density of test functions in the space of distributions — a clarification

Let $U \subseteq \mathbb{R}^n$ be open and denote by $\mathcal{D}(U)$ the space of all compactly supported smooth functions $U \to \mathbb{R}$. Let $\mathcal{D}^\prime(U)$ be the space of all ...
0
votes
1answer
24 views

Lipschitz constant of a Matrix Valued Function

Consider the function $H(w) = \sum_{i=1}^n f(w^T x_i) x_i x_i^T $, where $w\in \mathbb{R}^d$, $\forall i$: $x_i \in \mathbb{R}^d$, and $f: \mathbb{R} \to [0,1]$. Further, know $|f'(y)| \leq 1$ for all ...
2
votes
1answer
34 views

Spectrum invariance under the passage to a sub Banach Algebra

Let $\mathcal{B}$ be a unital Banach Algebra, fix $A \in \mathcal{B}$. $\sigma_\mathcal{B}(A) = \{ \lambda \vert \lambda I - A \, not \,invertible \, in \, \mathcal{B}\}$ the specturm of $A$ in $\...