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).

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Let $X$ be a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

Some definitions- $X$ is said to be strictly convex or rotund if for all $x,y\in S(X),\ x\ne y$ we have $\left\lVert\frac{x+y}{2}\right\rVert<1$ A point $x_0\in B(X)$ is said to be a exposed point ...
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How to prove that the convergence of $(f_n)$ is uniform on compact sets?

I'm reading Theorem 0.9 in this lecture note. Below is my attempt where I got stuck at the end. Could you elaborate on how to finish the proof? Let $C$ be an open convex subset of a Banach space $X$....
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Showing that a sequence of compact operators converge uniformly to their pointwise limit

I am working out of Phillipe G. Ciarlet's Linear and Nonlinear Functional Analysis with Applications and am struggling with exercise 4.9-5. The problem is as follows: Let $G$ be a function in the ...
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$X$ is strictly convex (rotund) iff for all $x,y\in S(X)$ with $x\ne y$ we have $\lVert x+y\rVert <2$

A Banach Space $X$ is said to be strictly convex or rotund if for all $x,y\in X$ we have $\lVert x+y\rVert<\lVert x\rVert+\lVert y\rVert$ unless $x,y$ are multiple of each other. We have to prove ...
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Weak Convergence for a continuous function without compact support

Let $X$ be an open subset of $\mathbb R^n$ and let $\Omega$ be a relatively compact, open subset of $X$. Let $\{ \mu_n\}$ be a sequence of positive measures that converge weakly to the positive ...
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If $\mathcal{F}$ is pointwise bounded, then $\mathcal{F}$ is locally equi-Lipschitz and locally equi-bounded

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X$ be a Banach space and $\mathcal{...
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Weak convergence of positive operators in Hibert space

Let $A_n$ be a sequence of positive operators on the Hilbert space $\mathcal{H},$ weakly convergent to an operator $A$ (necessarily positive). Does it imply the strong convergence ? By the weak ...
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Is the convolution of $L^2$ functions continuous? [duplicate]

Is the answer to the following question positive? The convolution of two $L^2(\mathbb R)$ functions is continuous I briefly recall it here: Take $f$ and $g$ $\in L^2(\mathbb R)$, then I want to show ...
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Equivalent definitions of Poincare inequality

I can't seem to find this anywhere but I can find that there are (at least) two definitions of the Poincare inequality. One is $$ \int_\Omega |f|^2 dx \leq c\int_\Omega |\nabla f(x)|^2 dx $$ for $f \...
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Uniformly convexity of a space

Let function $ \delta(\varepsilon)=\inf\{1 -\frac{\|x+y\|}{2}:x,y\in B_{1},\|x-y\|\geq \varepsilon \} $ be a modulus of convexity of Banach space $X$. How to prove that the space is uniformly convex ...
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Does my proof generalize this exercise about continuity of convex lower semi-continuous function?

I'm trying to solve below question (Proposition 0.7.) in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is ...
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Fredholm alternative vs variational methods

I was reading an exercise in a lecture notes of variational methods in order to show that if $\rho: \overline{\Omega} \rightarrow \mathbb{R}$ is continuous in a bounded domain $\Omega \subset \mathbb{...
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Almost closed forms

Suppose we have a closed (compact, without boundary) manifold $M$. Let's assume that it is orientable, although it might play no role in the question. Now, De Rham cohomology measures how far a closed ...
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If $g$ is $K$-lipschitz, show that constant function $g_{n}$ satisfies $ \|g-g_{n}\|_{L^{2}} \leq C(d) K \text{Vol}(\Omega) 2^{-p}$.

Let $\Omega$ be a domain that's covered with a finite number of dyadic cubes, i.e. cubes of the form $$ \prod_{i=1}^d\left[\frac{k_i}{2^p}, \frac{k_i + 1}{2^p} \right]. $$ Let $g \in L^{2}(\Omega)$. ...
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Stieltjes integral and Riesz representation theorem

Would you please explain how I can derive eq(1.2) from eq (1.1) by using Stieltjes integral in the Riesz representation theorem?? And is it necessary to suppose that the time interval begins from -$\...
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Almost sure convergence for lipschitz functions

Let $x_n \to x$($x_n$ sequence of random variables) s.t $\sum \mathbb{E}|f(x_n) - f(x)| < \infty$. For any $f$ Lipschitz and bounded. Then $x_n \to x$ almost sure. My attempt: As series converge, ...
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Apply Arzela-Ascoli theorem to unifomly bounded sequence in $H^1$

Let $\{u_{i}\}$ be sequence of smooth function defined on $\Bbb{R}^n$ such that $\|u_i\|_{L^2(\Bbb{R}^n)}$ is uniformly bounded in $i$ and $\|\nabla u_{i} \|_{L^{^2}(\Bbb{R}^n)}$ is also uniformly ...
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Finding metric projection mapping

Let $X$ = $c_0$, with supremum norm and $W$ = $\{ \{x_n\}_{n \geq 1} \in X$ $|$ $x_1=0\}$. Then how to find $P_W(x)$. Note that, $P_W(x)$ = $\{y_0 \in W : ||x-y_0|| = \inf ||x-y||,$ $ y \in W \}$. In ...
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unbounded self-adjoint operator approximation by bounded operators?

Consider an unbounded, self-adjoint operator $(T,D(T))$, on a separable Hilbert space $H$. We try to approximate $T$ by its truncation, that is, $T_{n}=P_{n}TP_{n}$, where $P_{n}$ is the projection ...
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Is the trace of this operator finite? [duplicate]

Let $H$ Hilbert space and let $Q\colon H \to H$ be a linear, self-adjoint, positive, trace-class operator. Let $X\colon H \to H$ be a linear, self-adjoint, positive operator. Does it follow then that $...
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Continous function a on convex set [closed]

Let $f:\Omega\rightarrow \mathbb R$ be a continous function where $\Omega$ is convex set. Does that mean that $f(\Omega)$ also a convex set?
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Fourier transform of a class of interesting functions to optimize a numerical algorithm

I try to speed up a numerical algorithm and I came to a class of real functions where I need the Fourier transform or the coefficients of the Fourier series with a large interval of them. The class of ...
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In a normed vector space, the convex function $f:C \to \mathbb R$ is locally Lipschitz if and only if $f$ is upper bounded on an open subset of $C$

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $(X, \| \cdot\|)$ be a normed ...
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Sets known with Hausdorff distance

Let A and B be two arbitrary compact sets. Then what property should hold by A and B such that the distance D(A, B) = D(B, A) = h(A, B), where D is the set distance and h is the Hausdorff distance. ...
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The functionals $\|\cdot\|_{L^{p,q}}$ do not satisfy the triangle inequality.

Given a measurable function $f$ on a measure space $(X,\mu)$ and $0<p,q\leq \infty$, define $$\|f\|_{L^{p,q}}=\left\{ \begin{array}{ll} \displaystyle{\left(\int_{0}^\infty\left(t^{1/p}f^*(t)\right)...
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maximal eigenvalue of self-adjoint operator is non-degenerate

I want some help in this one, if someone can prove or disprove it: "If $T$ is a compact, self-adjoint operator with positive spectral gap, then $||T||_2$ is always an eigenvalue and the ...
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Is this integral convolution continuous?

The following question is related to the one in Is this convolution continuous? Let $0 \leq \xi \leq T$, $Y_0 \in L^2([-T, T])$, $a_1 \in L^2([0,T])$ and consider the following function: $$ Y_{1}(\xi)...
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2 votes
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Does weak* convergence together with convergence of norms implies strong convergence in l1 as dual of c0?

Let $x^{*}_{n}$ is weak$^*$ convergent to $x^*$ in weak$^*$ topology on $l_{1}$ induced by $c_{0}$ and $\|x^{*}_{n}\|\rightarrow \|x^{*}\|.$ It is true that then we have $\|x^{*}_{n}-x^{*}\|\...
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Bounded operator has closed image if and only if there exists positive a constant $c$ such that $c \| x \| \leq \| Tx \|$.

In my notes I have the following theorem If $T:X \to Y$ is a bounded operator where $X,Y$ are complex Hilbert spaces. Prove that $Im (T)$ is a closed set if and only if there exists positive constant ...
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Under what minimum assumptions a null set is necessarily meager?

$A\subset \Bbb{R}$ is a null set if $\lambda(A) =0$ ($\lambda$ :Lebesgue measure) $A\subset \Bbb{R}$ meager if $A$ is countable union of nowhere dense sets (sets whose closure contains no nonempty ...
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Show that $\|v\|=\sup \{|\langle v,w \rangle|, \|w\|\leq 1\}$

I need some help. Let $(H,\langle\cdot,\cdot\rangle)$ a Hilbert space, and for every $v\in H$, defines a continuous function $\langle v,\cdot \rangle:H\to \mathbb{R}$. Show that for every $v\in H$: $$\...
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Extending a function that gives a value to convex functions to a measure

I am wondering if such a result exists (or similar) and or if there is a "simple" proof. Let $\mathcal X$ be a bounded and closed subset of a topological vector space, let $\Sigma$ be the ...
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A sequence of harmonic functions on $\mathbb{R}^{2}$ converging in distribution must necessarily converge locally uniformly to an harmonic function.

I am self studying the Rudin's book of Functional Analysis and I stumbled upon this problem. Given a sequence $\{ f_{j} \}_{j}$ of harmonic functions on an open set $\Omega$ of $\mathbb{R}^{2},$ if $\{...
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Modulus of convexity - continuous function

I need to prove that modulus of convexity of a Banach space is a continuous function. Where can I find this proof? Modulus of convexity we define in the following way: $$ \delta(\varepsilon)=\inf\{1 -\...
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Bounded from above convex function on a normed vector space is locally Lipschitz

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) I'm trying to generalize this result to ...
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1 vote
1 answer
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Exercise 4.19 (1) of Brezis

I am trying to solve the following exercise of Brezis' book on Functional Analysis. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence in $L^p(\Omega)$ with $1 < p < \infty$ and let $f \in L^p(\Omega)$...
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Fractional powers of the Laplacian

I have recently started to study fractional fractional powers of the Laplacian. In the books I've read, fractional powers are defined only for $$-\Delta =-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}....
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3 votes
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$T*\varphi=\sin(x)$

I am looking for a distribution $T:\mathcal D(\mathbb R)\to\mathbb C$ and a test function $\varphi\in C_c^{\infty}(\mathbb R)$ with $$T*\varphi = sin(x),$$ but I can't think of any pair $(T,\varphi)$ ...
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How many points does a line intersect a sphere in an infinite-dimensional normed vector space?

Let $(E, |\cdot|)$ be a n.v.s. We fix $r>0$ and $x,y \in B(0, r)$ such that $x\neq y$. Here $B(0, r)$ is the open ball centered at the origin and having radius $r$. The set of all points in the ...
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Is a continuous function on compact convex set where the boundary is mapped to the set a self mapping?

Let $K ⊂ R^n$ be convex and compact with $0$ in the interior of $K$. Let $f ∈ C(K, R^n)$ with $f(∂K) ⊂ K$. If this is the case, do we in fact have $f(K) \subset K$. It is probably not the case that ...
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The miraculous nature of the "matrix coefficients" $\langle Tv,v \rangle$ (especially in the context of positive type functions)

$\newcommand{\ak}[1]{\langle #1 \rangle}$I've noticed that for linear operators $T$ and an inner product $ \ak{\bullet, \bullet }$, the expression $\ak{ Tv,v}$ tends to show up a lot. For instance, it ...
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Is $A\subset X$ bounded when $\{\Delta(x):x\in A\}$ is bounded for all $\Delta\in X^*$, $X$ is a normed space and $A\subset X$? [closed]

Let $X$ a normed space and $A\subset X$. Suppose that for all $\Delta\in X^*$, $\{\Delta(x):x\in A\}$ is bounded. Prove that A is bounded
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1 answer
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Connectedness of Toeplitz operator spectrums

Im working through the following theorem, and I don't understand the explanation. Theorem: Let $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ be the Toeplitz operator with $T \subset \mathbb{C}$ the ...
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2 answers
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Characterizations for normal distribution and Poisson distribution

Let $X$ be a real valued random variable such that for all $f \in C_c^{ \infty}( \mathbb R)$ we have $ \mathsf E(Xf(X))= \mathsf E(f'(X))$. Show that $X$ has the standard normal distribution. Let $ \...
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Deficiency indices of the Laplace operator on the unit disk

Let $\mathbb{D}=\{(x,y)\in \mathbb R^2: x^2+y^2\le 1\}$ be the unit disk. Let consider the Laplace operator $\Delta_0$ defined on $C_0^\infty(\mathbb D)$, the space of smooth functions $\mathbb D\to \...
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Reference for time dependent traces

Consider the spaces $H^{1/2}(0,T; L^2(\partial\Omega))$, or $L^2(0,T;H^{3/2}(\Omega))$ and what not. I'm interested in a reference book illustrating the meaning, properties of these spaces (so, ...
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Flood Hydrograph Characteristics with Delta Function

I am trying to understand part of my lecture notes on solving a simple model for a river in the case of a flash flood. We have the equation, $$\frac{\partial A}{\partial t} + cA^m \frac{\partial A}{\...
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-3 votes
1 answer
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Orthogonal complement relationship [closed]

Suppose two spaces are orthogonal: $A\perp B$. Is $A^\perp \perp B^\perp$?
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1 vote
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Does this implies $\mathcal{M}[T]_{\beta_1 \beta_2}$ is similar to the $\mathcal{M}[T]_{\beta_3 \beta_4}$?

$T\in\mathcal{L}(V) $ where $\dim(V) <\infty$ Consider $\beta_1, \beta_2, \beta_3, \beta_4$ four bases of $V$ . Does this implies the matrix $[T]_{\beta_1 }^{\beta_2}$ is similar to the matrix $[T]...
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Which one should I start studying - Lie Groups/Algebras or Functional Analysis or Differential Geometry? [closed]

I major in Physics(an excuse for sloppy math practices) and wish to go deeper in terms of understanding the concepts and improving mathematical rigorousness. I wish to study Algebraic Topology, ...
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