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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A minimisation problem in an Hausdorff space

Let X an hausdorff space, K a compact set, f a lower continuous function. How can you prove that f has a minimum on K? Ps: be careful X is not specially a metric space
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As $\lambda \to \infty$, $||R_{\lambda}(T)|| \to 0$

"Let $T \in B(X,X)$. Prove that $||R_{\lambda}(T)|| \to 0$ as $\lambda \to \infty$." We have that $R_{\lambda}(T)x=(T-\lambda I)^{-1}(x)$. The problem doesn't specify but the book says that they will ...
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11 views

Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
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31 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
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1answer
14 views

Topology induced by semi-norm and P-topology

Let V be an abstract vector space over F and P a family of semi-norms on V. We can define a topology on V by declaring its base to consist of subsets of V of the form {v∈V | p1(v−x)<ϵ ∧ … ∧ ...
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16 views

Proving dense set is core for a self adjoint operator

Let $A$ be a self adjoint operator in a Hilbert space $H$ and $D\subseteq D(A)$ a dense subset such that $$ e^{iAt}:D \to D. $$ How can I show that $D$ is a core for $A$? I need to show that ...
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a problem about the application of Banach-Steinhaus theorem [on hold]

Let $\{x_k\}$ to be a sequence in Banach space $X$. Prove that if for any f in $X^*$, $\sum_{k=1}^\infty |f(x_k)|<+\infty$, then there exist a constant M s.t. for every $f\in X^*$ we have ...
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0answers
14 views

Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
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1answer
44 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition, a nonexample and a comparison see: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample ...
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1answer
24 views

Approximation of $f \in L^1_{loc}$

I am trying to prove the following statement: If $\Omega$ open in $\mathbb{R}^n$, $f \in L^1_{loc}(\Omega)$ (a set of all functions whose integrals on compact sets exist) and $\int_{\Omega}f\cdot g ...
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8 views

Visual notion of tangential gradient

Before I begin, this question is related to personal reading and is not in any way connected to an assessment/assignment. I am struggling to visualise the tangential gradient. As I understand it, the ...
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1answer
8 views

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$.

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$. Normally, I would do is find a matrix representation and ...
2
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1answer
26 views

Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$.

Let $H=L^2(a,b)$ with $a<b$. Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$. Verify also that $$\{(b_1-a_1)^{-\frac{1}{2}} \cdot \dots ...
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2answers
23 views

Norm of orthogonal projection operator $P$, if $\text{Im}\,P\subseteq\text{Im}\,Q$, with $Q$ also an orthogonal projection

In Rynne & Youngson: Linear Functional Analysis, there is an exercise stated as Let $\mathcal{H}$ be a complex Hilbert space and let $P$, $Q\in B(\mathcal{H})$ be orthogonal projections. Show ...
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0answers
16 views

$[f=q]$ is closed implies $f$ is continuous. [on hold]

Define $H=[f=q]$ to be the set of points such that $f(x)=q$. Let us assume that H is closed. Then the complement of H is open and nonempty (since $f$ does not vanish identically - ""i don't understand ...
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2answers
30 views

Weak topology : defined by linear mapping vs semi-norm

Wikipédia: -The weak topology on X is the initial topology with respect to X* (let's note it T') -If the field K has an absolute value , then the weak topology σ(X,F) is induced by the family of ...
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2answers
27 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
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1answer
29 views

Determine adjoint operator

Let $K\in D\{(x,\xi)\in \mathbb {R}^2 : x > 0, \xi > 0\} $ and $L (\phi)(x)=\int_0^x K (x,\xi)\phi (\xi)d\xi $ for $\phi \in D (\mathbb {R})$ $D $ is the space of testfunctions I know that ...
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1answer
32 views

How prove this indentity$\langle F'',g\rangle=-\frac{1}{4}\int_{0}^{+\infty}\frac{g(x)-g(0)}{x^{3/2}}dx$

For the generalized function F defined as $\langle F,g\rangle =\int_{0}^{\infty}\sqrt{x}g(x)dx$,show this the following equalities $$\langle F'',g\rangle ...
4
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1answer
46 views

Fourier transform inversion formula for $f\in L_1(\mathbb{R}^n)$ and Dini condition

Let us define the Dini condition for a function $f\in L_1(-\infty,\infty)$, i.e. Lebesgue summable on $\mathbb{R}$, as Given an $x\in\mathbb{R}$ there is a $\delta>0$ such that the Lebesgue ...
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0answers
52 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
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2answers
35 views

Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
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0answers
26 views

Subset being orthonormal basis if $\sum_{n\geq 1} \|a_{n}-b_{n}\|^{2} < \infty$ [on hold]

I have a question Supposing $X$ is a Hilbert space and lets suppose that $A=\{a_{n} \ : \ n\geq 1\}$ is an orthonormal basis of $X$. Let $B=\{b_{n} \ : \ n\geq 1 \}$ be an orthonormal subset of $X$. ...
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0answers
24 views

Is there are “sphere” associated to any topological vector space?

If I have a topological vector space that is not locally compact, is it still possible to associate to it some natural "sphere" like object? For locally compact Hausdorff spaces, the my first guess ...
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0answers
36 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
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2answers
18 views

Find a representation of the functions $f(x)=1-x$ and $g(x)=\chi_{(0,\frac{1}{2})}(x)$ in terms of the exponential basis $\{e^{2\pi ixn}\}$

Find a representation of the functions $f(x)=1-x$ and $g(x)=\chi_{(0,\frac{1}{2})}(x)$ in terms of the exponential basis $\{e^{2\pi ixn}\}$ of $L^2(0,1)$. Recall that the characteristic function ...
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0answers
18 views

Approximating $v \in W^{1,p}_0(\Omega)$

Let $p > 1$. Let $V=W^{1,p}_0(\Omega)$ and $H=L^2(\Omega)$. Suppose $\{\lambda_i\}$ is a basis in $V \cap H$ which is smooth, and orthonormal in $L^2$. Given $v \in V$, is it possible to find a ...
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1answer
43 views

Show that the set $\{v_n\} \subset l^2$, $v_j$ is orthonormal.

Show that the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even is orthonormal. Is it a Schauder basis of ...
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2answers
25 views

Parseval's Identity holds for all $x\in H$ implies $H$ is a Schauder basis

Prove that any set $\{v_j\}_{j \in \mathbb{Z}}$ for which the Parseval identity $\|x\|^2=\sum_{j=1}^\infty |\langle v_j,x\rangle|^2$ holds for every $x \in H$ is a Schauder basis. I know that a ...
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0answers
42 views

Question about proof in functional analysis book

I'm currently working through introductory functional analysis from kreyszig, and I don't quite understand one of the proofs. this is it: My question is why it is necessary to go from $k = 1, 2, ...
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1answer
31 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
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1answer
17 views

Prove that the image of an orthonormal basis through a linear, invertible and bounded transformation is a bounded and unconditional Schauder basis.

Prove that the image of an orthonormal basis through a linear, invertible and bounded transformation is a bounded and unconditional Schauder basis. I am having trouble finding a starting point for ...
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2answers
28 views

Is every Lebesgue measurable function bounded on a set of positive measure

Let $f$ be a Lebesgue measurable function from $[0,1]\to\mathbb{R}$. Let $\mu$ be Lebesgue measure. Does there exist a measurable set $B$ with $\mu(B)>0$ and an $M>0$ such that for all $x\in B$, ...
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3answers
24 views

Product of two positive compact, self adjoint operators

If we have two positive compact , self adjoint operators; $A$, $B$. Is the product $AB$ a positive operator?
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1answer
39 views

A bounded linear operator on $L^{2}$

I have a quesiton about a bounded linear operator and so on. Let $(E,\mathcal{B},m)$ be a $\sigma$-finite measure space and $G$ be a bounded linear operator on $L^{2}(E;m)$. We assume the follwing ...
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1answer
19 views

Spectral Measures: Subspace Characterization

Disclaimer This thread is related to: Spectral Measures: Subspace Decomposition It is meant to record. See: Answer own Question It is written as question. Have fun! :) Question Given a Hilbert ...
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1answer
32 views

Weak Convergence and its Relationship to a Sequence of Norms

"Prove that $x_n\xrightarrow{w} x$ implies lim inf$||x_n|| \geq ||x||$." I'm trying to understand weak convergence better through this exercise. Here, $\xrightarrow{w}$ means weakly convergent, i.e. ...
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1answer
24 views

Functional analysis question about proving $J(\phi)=||\phi||$

Let $M$ be a closed subspace of a normed space $X$ and let $Q$ be the quotient map i.e. $Qx=x+M$ for all $x \in X$. Let $M^{\perp}=$ {$ \psi \in X^{*} : \psi(m)=0$ $\forall m \in M$}. Prove that : ...
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3answers
32 views

Spectrum of Multiplication Operator $T$ in $C[0,1]$

"Let $X=C[0,1]$ and $v \in X$ be a fixed function. Let $T$ be the multiplication operator by $v$, i.e. $Tx(t)=v(t)x(t)$. Find the spectrum of $T$." This is an exercise from a PDF of notes I found ...
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14 views

Extension theorem for locally Lipschitz functions

Let A be a subset of a metric space $(X,d)$ and f be a real valued locally lipschitz function on A. Does there exists a real valued locally lipschitz function on X which is an extension of f? or under ...
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1answer
42 views

Check if $f_n(x)=x^n-x^{2n}$ is convergent in C([0,1]) [on hold]

Check if $f_n(x)=x^n-x^{2n}$ is convergent in C([0,1]) - (continuous functions)
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1answer
30 views

then $g$=$f$ in a hilbert space [on hold]

Let H be a Hilbert space. Let $f$,$g$ $\in$ $H'$ with ker $f$ = ker $g$ with $||f||$ =||g||: Let g(x) = f(x) for some non-zero $x$ $\in$ $kerf \perp$. Then g = f:
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9 views

Power series of bounded linear operators

If $f$ is a complex analytic function, one can define a matrix function $F$ using the Taylor series of $f$ by $$ F(A) = f(0) + f'(0)\cdot x + f''(0)\cdot \frac{A^2}{2!} + \cdots $$ If the radius of ...
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1answer
18 views

Show that $u(x)=\ln\left(\ln\left(\frac{1}{1+|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$.

The entire problem statement is: Let $n>1$ and let $U=B(0,1)\subset\mathbb{R}^n$. Show that $u:U\to\mathbb{R}$ given by $$u(x)=\ln\left(\ln\left(\frac{1}{1+|x|}\right)\right)$$ is in ...
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0answers
35 views

Is it possible to prove continuity without nets?

In the proof of the following theorem: Let $X$ be a compact Hausdorff space. Then $\varphi : X \to \Omega (C(X))$ given by $x \mapsto e_x$ where $e_x$ is evaluation at $x$ and $\Omega$ is the ...
3
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1answer
46 views

Adjoint of completely continuous operator is completely continuous

In the proof of the fact that the adjoint operator $A^\ast$ of a completely continuous linear operator $A:E\to E$ mapping a Banach space into itself is also completely continuous on $E^\ast$ endowed ...
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1answer
38 views

Spectrum of a Self-Adjoint Operator is Real

Preparing for an exam in functional analysis, I'm trying to show that for a self-adjoint operator $A$, $\sigma(A) \subset \mathbb{R}$. I came across the following proof in the book (or rather, lecture ...
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0answers
19 views

When is the spectra of a function composed completely of eigenvalues?

For a linear operator $T$, define its spectrum to be $\sigma(T)$. Also define $L = \{\lambda \mid Tv = \lambda v$ for some $v\in B$. Certainly, $L \subset \sigma(T)$, and if $T$ is finite ...
2
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0answers
36 views

baire category and the union of dyadic balls of rational center

Suppose that $\{r_n\}_{n=0}^\infty$ is an enumeration of $\mathbb{Q}^N$ and $U = \bigcup_{n=0}^\infty B(r_n,2^{-n})$. We can use a trivial measure theory argument to prove that $U \neq \mathbb{R}^N$. ...
0
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0answers
22 views

Prove quasiconvexity of a multivariate function [on hold]

I would like to prove that the following function is quasiconvex: $$ f(x_1,x_2) = -x_1*x_2 -1 $$ How do i prove its quasiconvexity (without augmented Hessian matrix) ?