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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107 views

$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some boundary conditions. Consider the weak form ...
7
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3answers
799 views

The difference between hermitian, symmetric and self adjoint operators.

I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have to go ...
8
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2answers
569 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
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1answer
47 views

Linear onto isometry.

Let $K$ and $L$ be two compact set and $T$ is an linear onto isometric from $C(K)$ to $C(L)$. My question is that $T(1)$ is the identity map in $C(L)$, where 1 is the identity map in $C(K)$ . give me ...
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1answer
82 views

The Levi-Civita connection in infinite dimensions

Is there an analogue of the Fundamental Theorem of Riemannian Geometry for (some subclass of) infinite-dimensional manifolds?
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1answer
62 views

“Duality” for weak $L^p$ spaces

Let $1<p<\infty$. Denote by $L^{p,\infty}$ the weak $L^p$ space in $\mathbb{R}^n$ and let $f\in L^{p,\infty}$ where we define the weak $L^p$ quasinorm as $$\|f\|_{p,\infty} = \sup_{\lambda ...
2
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2answers
99 views

Which is the relationship between weak convergence and pointwise convergence?

In one of my indipendent works at functional analysis course have to come up with an explicit way of telling which is the relationshpip between weak and pointwise convergence for $C(K)$ where $K$ is a ...
2
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1answer
79 views

Link between Chebyshev polynomials and best approximants

I'm reading Interpolation and Approximation by Davis, more specifically "Best Approximation" Chapter VII. Let $n \in \mathbb N$. Let $C[a,b]$ denote the set of continuous real functions over ...
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1answer
326 views

why compact support implies a function vanished at boundaries?

"A function has compact support if its support is a compact set." While support of a function $u:G\rightarrow\mathbb{R}$ is defined to be $supp(u)=\overline{\{x:G|u(x)\neq0\}}$ But lately, Another ...
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0answers
25 views

Existence of the solution of a linear parabolic pde

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in ...
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0answers
26 views

Embedding to $L^\alpha(0,T;L^\beta(\Omega))$

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the space $W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$. It is well-known that $W \subset C([0,T];H)$ where $H = ...
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1answer
430 views

Condition for degenerate eigenvalues for a matrix

Given a diagonalizable matrix $M$ (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ...
2
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1answer
93 views

Proving functions are uniformly continuous help!

a) Prove that $f(x)=x^{1/4}$ is uniformly continuous on $[0,\infty)$. Show that this method can be extended inductively to any $f(x)=x^{1/p}$ for any $p=2^n$ b) Prove directly from the definition ...
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1answer
51 views

Want to prove that the Hilbert transform of a $C^1(\mathbb T)$ function is the principal value of the convolution with $\cot(\pi x)$

So here is my problem, Let $L^2_0:=\{f\in L^2: \hat{f}(0)=0\}$ and consider the Hilbert transform given by the following map $$H:L^2_0([0,1])\rightarrow L^2_0([0,1])$$ $$f\mapsto (\mathcal ...
2
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1answer
76 views

Weak and Norm convergence in Banach Space

I know (and have proven) that in a Hilbert space, $\mathscr{H}$, if a sequence $z_i\overset{w}{\to}z$ and $\|z_i\|\to\|z\|$, then $\|z_i-z\|\to0$. I'm trying to find a counterexample in a Banach ...
4
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133 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
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3answers
39 views

Some questions of vectors and dense subsets

I have a couple of quick functional analysis related questions: 1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. ...
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34 views

Lipschitz function and uniform boundedness principle

Let $(S,d)$ be a metric space and $X$ be a normed space. Show that if $f:S\to X$ is a function such that for all $L\in X^*$, $LOf:S\to {\Bbb F}$ is lipschitz(there is a constant $M>0$ such that for ...
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1answer
43 views

what are contraction(Lipschitz) maps on $\mathbb C$?

We say a map $f:\mathbb C \to \mathbb C$ is contraction(Lipschitz) if $|f(z_{1})- f(z_{2})| \leq C |z_{1}- z_{2}|$ for every $z_{1}, z_{2} \in \mathbb C$ and $C$ is some constant. Trivial Examples: ...
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1answer
102 views

Probability Density Function of non decreasing function

Can anyone please help me with this random variable question I've stumbled across. Recall from calculus that a function $h$ is called non-decreasing if $x\leq y$ implies $h(x)\leq h(y)$, for every ...
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0answers
61 views

Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
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1answer
88 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
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1answer
258 views

Prove that weak convergence does not necessarily imply strong convergence without counterexample.

Here is the set of original problems. Let $\{x_n\}$ be a sequence in a normed linear space $X$. Prove that: Strong convergence implies weak convergence with the same limit. The converse ...
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1answer
56 views

In the proof of every isomorphism of $\mathbb{C}^n$ onto an $n$-dimensional subspace of a complex topological vector space is a homeomorphism

I was reading the proof of the following theorem in Rudin 2/e: Theorem 1.21 If $n$ is a positive integer and $Y$ is an $n$-dimensional subspace of a complex topological vector space $X$, then ...
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1answer
27 views

($\forall x\in E_1 \exists y\in E_2: f_1(x_1)=f_2(x_2))\ \Longrightarrow \ x \mapsto y$ continuous

I'm reading through some notes on functional analysis where on a couple of occasion we where in the following setting: $E_1$ and $E_2$ are two Frechet spaces and $f_i:E_i \rightarrow F$ two ...
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0answers
31 views

Properties of this set of functionals (mixed pairings)

(from the 4th page of http://www.math.toronto.edu/mccann/papers/econ.pdf) Let $X$ be a compact Hausdorff space, and let $\omega$ be a Borel probability measure on $X$. A Borel probability measure, ...
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51 views

How to make sense of the Fourier transform of this distribution

I want to compute the Fourier transform of this distribution: $$D(f)=\int_{\mathbb{R}} f(t,t^2) \frac{dt}{t}$$ ($f$ a Schwartz function on $\mathbb{R}^2$, the integral interpreted with a Cauchy ...
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2answers
115 views

Want to show that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1-periodic $L^2$ functions.

So here is my problem, I would like to prove that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1- periodic $L^2([0,1])$ functions with respect to, $$\langle ...
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2answers
134 views

strict subadditivity of the norm in uniformly convex Banach Spaces.

So here is my question, I would like to prove the following, Let $(X,||\cdot||)$ be a uniformly convex Banach Space. Then the norm is strictly subadditive i.e $\forall x,y\in ...
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1answer
53 views

I have question about the operator space theory

I have question about the operator space theory. How to solve equation (2.2.1) in details
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1answer
52 views

Fréchet derivative, is this true?

I was just wondering whether the following statement is true: Let $H_1,H_2$ be Hilbert spaces and $\{e_n\}_{n\geq 0}$ be an orthonormal basis of $H_2$. Let $f:H_1\rightarrow H_2$ be an operator (not ...
2
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1answer
43 views

A linearly independent set about approximate units

Let $B$ be a C*-algebra and $\{b_{i}\}_{i=1}^{n}\subset B$ be linearly independent. If we take $\{f_{k}\}\subset B$ which is approximate units, then can we find a large $k$, such that $\{b_{1}f_{k}, ...
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187 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
0
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1answer
18 views

A symbol of commuting ranges in tensor product

Here is a proposition of tensor product: ($A,~B,~C$ are C*-algebras) Proposition 3.1.17 Given two *-homomorphisms $\pi_{A}: A\rightarrow C$ and $\pi: B\rightarrow C$ with commuting ranges (i.e., ...
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3answers
60 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
0
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1answer
56 views

Show that a sequence of functions is not compact in $(C^1([-1,1]),\lVert\,\cdot\,\rVert_{\infty})$

Let $F:=(f_n)_{n\in\mathbb{N}}$ where $$\forall x\in[-1,1]\,\,\,\,\,\,\,\,\,f_n(x):=\mid x\mid^{1+\frac{1}{n}}$$ I have to prove that $F$ is not compact in ...
2
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1answer
90 views

Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
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1answer
27 views

What is this Space called?

We say $\mathcal H_2$ is a space contains all stochastic processes $F$ which satisfy: (i) $(F_t, t \in [0,T])$ is adapted (ii)$||F||_2 < \infty$ so what does this space H2 called here? does it ...
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1answer
34 views

Normalized States

A linear functional is normalized iff it preserves identity: $$\|\omega\|=1 \iff \omega(\mathrm{id})=1$$ Can somebody help me proving it? (I just remember it was kind of an easy thing.)
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37 views

A question about the quotient isomorphism

Let $X$, $Y$ be two vector spaces and $f: X \rightarrow Y$ be a surjective map. If $M\subset $ker$f\subset X$ is a subspace, and $X/M$ is isomorphic to $Y$ (it is induce by $f$), can we conclude that ...
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99 views

Dilation and translation of the Dirac Delta distribution

Given $\delta(ax) = \frac{1}{|a|}\delta(x)$ and $\delta(ax-b) = \frac{1}{|a|}\delta(x-b/a)$ , it it true also that $\delta(a(x-b)) = \frac{1}{|a|}\delta(x-b)$ ?
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44 views

Positive function approximation

Let $(X, \mathscr{S}, \mu)$ and $(Y, \mathscr{T}, \lambda)$ be two $\sigma$-finite measure spaces, give $X\times Y$ the product measure, then is it true that for any positive measurable function ...
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1answer
51 views

Looking for a “trace inequality” of normal derivative in $L^1$

Let $U = \Omega \times (0,\infty)$ with $\partial U = \Omega \times \{0\}$. Let $w \in H^1(U)$ with $(w|_{\partial U})^{\frac 1 m} \in L^1(\partial U)$. I am looking for a trace inequality of the ...
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2answers
63 views

Why is $L^p$ isomorphic to $(L^p)^2$

Is it possible to say why the spaces in the title are isomorphic as Banach spaces? Is their a Theorem that says this or is it even possible to find an explicit representation of this isomorphism?
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1answer
47 views

The multiplication of tensor product

Proposition 3.1.15 (Multiplication). Let $A$, $B$ be C*-algebra, the tensor product $A\odot B$ (denotes the algebraic tensor product) has a multiplication defined by $$(\sum\limits_{i}a_{i}\otimes ...
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1answer
42 views

The involution of tensor product

Proposition 3.1.8 (Linear independence). If $\{x_{1},...,x_{n}\}\subset X$ are linearly independent, $\{y_{1},...,y_{n}\}\subset Y$ are arbitrary and $$0=\sum\limits_{i=1}^{n}x_{i}\otimes y_{i}\in ...
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1answer
48 views

Fourier transform and sufficient condition…

Does anyone could give me a sufficient condition on $f$ so that the Fourier transform of $f$ (denoted as $\hat{f}$) is in $L^{1}(\mathbb R)$. The Fourier transform here is the linear operator ...
1
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1answer
78 views

Closed Graph Theorem

Let $(x_n)$ be a Schauder basis of $X$ and $(y_n)$ an equivalent one to $Y$. They are supposed to be equivalent, hence for every sequence $(a_n)$ the series $\sum_{n \in \mathbb{N}} a_nx_n$ converges ...
1
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0answers
36 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
2
votes
1answer
91 views

Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...