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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Need example of: Algebraic sum of closed vector subspaces need not be closed

I've read somewhere that given two closed subspaces $V_1,V_2$ in topological vector space $X$, their algebraic span $V_1+V_2=\{x_1+x_2 |x_i \in V_i, i=1,2\}$ need not be closed. I always thought that ...
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1answer
37 views

Closed convex hull of unitaries

If a C*-algebra ${\cal U}$ contains a non-unitary isometry $S$, show that $$\|S-A\|>\frac{1}{2n}$$ for every $A=\sum_{i=1}^n \lambda_iU_i$ which is the convex combination of $n$ unitaries. Thanks ...
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1answer
38 views

What is the importance that an assumption needs to state whether a space is Banach space?

I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition. For example, going ...
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1answer
44 views

proof of existence of a solution with $ f \in L^1$

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in L^1(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ for the problem ...
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1answer
41 views

Existence of unbounded operators on Banach spaces

I'm confused by the questions Discontinuous linear functional and Example of an unbounded operator which ask about unbounded linear functionals/operators on Banach spaces. I don't understand how ...
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2answers
60 views

Show linear map is continuous

Consider the linear map $T : \mathbb{R}^3 \to \mathbb{R}^3$ as: $T(x_1, x_2, x_3) = (x_1+x_3, x_2-x_1, x_3)$ I know that every linear map is contiuous if the vector space $X$ is finite dimensional. ...
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2answers
64 views

Can someone give me a hint on how to solve this question

Let $(X,d)$ be a compact metric space. For each $n \in \mathbb{N}$ we have $ f_n:X \to \mathbb{R}$ be a continuous function such that $f_n(x) \geq0 \forall x \in X$ . Assume that for all $x \in X$ the ...
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1answer
39 views

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma

Here's Theorem 4.3-2 (i.e. the Hahn Banach theorem for normed spaces): Let $f$ be a bounded linear functional defined on a subspace $Z$ of a normed space $X$. Then there exists a bounbed linear ...
3
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1answer
58 views

Is it incorrect to say that a functional “maps functions to numbers”?

Does a "functional" always takes in a function and spit out a number? This is what a professor said in class a long time ago but now I am studying Frechet derivative and a claim was made that a ...
2
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1answer
54 views

Continuity at $x=0$ of this function

Not a hard exercise:$$f(x)=\frac{1}{x^3}\cdot \int_{-x}^x \sin(4t^2) \, \text{d}t \quad \text{where} \space x\ne 0\:$$ $$f(x)=5\:;\:x=0\:$$ Checking it's continuity at $x=0$ by using L'Hospital's ...
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0answers
24 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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1answer
52 views

Domain Issue: Notation

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ It is well known that:* $$A=A^{**}\iff ...
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0answers
43 views

Uniform convergence of functions involving normal CDF

Consider two sequences of continuous functions $(f_n)$ and $(g_n)$ for $n \geq 0$ defined by $$ f_n (x) := \int_0 ^t \Phi\left(\frac{x\Phi ^{-1}(\alpha(s) + \beta_n(s))+\Phi^{-1} ...
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0answers
19 views

$N$-dimensional linear operator is normal, Lagrange interpolation?

Is there a way to see that an $N$-dimensional linear operator $A$ is normal if and only if $A^\dagger$ can be represented as a linear combination of $I, A, A^2, \dots, A^{N-1}$ using Lagrange ...
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0answers
15 views

Fredholm determinant for general kernel (discontinuous function, distribution, etc.)

For a Fredholm equation of the second kind $\phi(x) - \lambda \int_a^b k(x,y) \phi(y) dy = f(x)$, the solution can be obtained by constructing a solving kernel for any regular value of $\lambda$, for ...
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1answer
29 views

Wave Operators: Unitarity

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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2answers
39 views

Prove Weierstrass function has a pole of order 2 for all $\omega \in L$

I need some help to prove Weierstrass function has a pole of order 2. The Weierstrass function $\wp$-function of lattice $L$ is defined by $$\wp(z) = \wp(z; L) = \frac{1}{z^2} + \sum_{w \in ...
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1answer
42 views

Is linear dual space a misleading term?

A linear dual space consists of all linear functionals that sends a function in the space $X$ to its underlying field But the linear space itself does not send element of the field to the space $X$, ...
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2answers
44 views

Dense Operators: Kernel

This thread is Q&A. Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a dense operator: ...
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1answer
37 views

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$ for all $n\in \mathbb{N}$. Let ${e_n}$ be a ...
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1answer
55 views

Boundedness of $A$ in the operator equation $Au = f$ of $-\Delta u(x)=f(x)$.

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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1answer
33 views

Partial Isometries: Positivity

Given a unital C*-algebra $1\in\mathcal{A}$. Then implication holds: $$J\in\mathcal{A}:\quad JJ^*J=J\implies\sigma(J)\geq0$$ How can I check this? (Operator-algebraically?)
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1answer
36 views

Dirichlet energy and Fourier transform

Is there a direct relationship between the Dirichlet energy of a function: $$E(f)=\int_{\Omega}\lvert\nabla f(\mathbf{x})\rvert^2\mathrm{d}V$$ and its Fourier transform ...
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4answers
88 views

Calculate function: $\int_{a}^{b} \left(f{(x)}\right)dx=c$

Is there a way to find the function $f{(x)}$ for a given value of $a,b,c$? $$\int_{a}^{b} \left(f{(x)}\right)dx=c$$ For example: $a=0,b=1,c=\frac{1}{3}$ we get: $$\int_{0}^{1} ...
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0answers
23 views

Compact embedding of Banach spaces; when does a bounded sequence converge in the bigger space?

Let $X$ and $Y$ be Banach spaces with $X$ compactly embedded in $Y$. What more assumptions do I need on the spaces to ensure that: every sequence $x_n \in X$ which is bounded ($\lVert x_n \rVert_X ...
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0answers
84 views

How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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1answer
49 views

Is this a correct perspective?

Consider I have a sensor which is measuring some disturbance and I am converting the disturbance into complex numbers at a regular rate. I have done it for some time and therefore I get a series of ...
1
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1answer
22 views

Wave Operators: Isometry

This thread is only Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
1
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0answers
21 views

Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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1answer
32 views

Isometries: Weak vs. Strong

Given a Hilbert space $\mathcal{H}$. Consider isometries: $$R_\lambda\in\mathcal{B}(\mathcal{H}):\quad R_\lambda^*R_\lambda=1$$ Then it follows: $$R_\lambda\rightharpoonup R\implies R_\lambda\to R$$ ...
2
votes
1answer
42 views

Is $B(H)$ the weak-$*$ closure of $K(H)$?

I am getting the following result: If $H$ is a Hilbert space, then the weak-$*$ closure of $K(H)$, the space of compact operators on $H$, is $B(H)$, the space of bounded operators on $H$. Is this ...
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1answer
35 views

Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$. What is a suitable $Y$ for this to hold?

Let $A$ and $B$ be bounded linear operators on a normed space $Y$ into $Y$. Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$ Here is what I have ...
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1answer
20 views

Passing complemented subspaces to duals

Sorry for this rather basic question from Banach space theory. Suppose I have a complemented subspace $E$ of a Banach space $X$. So let's write $i:E\to X$ to be the inclusion map. Then I have a ...
2
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1answer
72 views

Existence of specific weak derivative

Suppose there exits a sequence $(\phi_n)_n\subset C_0^\infty(\Omega)$, where $\Omega\subset\mathbb{R}^2$ is a bounded domain with $C^\infty$ boundary, such that $(\partial_1+\partial_2)\phi_n$ ...
2
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0answers
90 views
+100

Polar Decomposition: Adjoint

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\subseteq\mathcal{H}\to\mathcal{K}:\quad A=A^{**}$$ And its decompositions: ...
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1answer
67 views

Sobolev functions counterexample

Let $A=(0,1)^{d}$.Does anyone have a simple example of a funtion in $H_0^1(A)\cap H^2(A)$ that is not in $H^2_0(A)$? Thanks a lot.
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39 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F ...
4
votes
1answer
34 views

Why lower semicontinuity?

I'm reading a proof on the existence of a solution to a minimisation problem, but I'm stuck. I give a brief summary of the arguments up to the point at which I'm stuck(at the yellow box). ...
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1answer
30 views

Weyl sequence for closure of an operator

I'm trying to solve following exercise and need some hints. Let $A= \bar{ A_0 }$ be closure of $A_0$ - a densely defined operator. Suppose $f_n \in D(A)$ is Weyl sequence for $z \in \sigma (A)$. Show ...
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1answer
30 views

Measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)h(t)$ given measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)$?

Suppose I know that, given $f(t), g(t) \in L^2(\Omega(t))$, $$t \mapsto \int_{\Omega(t)}f(t)g(t)$$ is measurable on $([0,T], Lebesgue) \to (\mathbb{R}, Borel)$. Suppose $h(t) \in L^\infty(\Omega(t))$ ...
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1answer
71 views

Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...
0
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1answer
38 views

Is $C(\Omega) \cong\prod_{n \in \mathbb{N}} C(K_n)$?

Let $\Omega$ be an open set in a topological space and $C(\Omega)$ be the vector space of continuous complex valued functions with the topology given by the following family of seminorms: ...
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1answer
34 views

Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of ...
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1answer
41 views

Showing equivalence of seminorms

Let $K =[0,1]$ and let $X \subset C^{\infty}(K)$ be the subspace of all functions vanishing on the end points of $K$. Show that the following seminorms are equivalent: $||D^nf||_1$ $||D^nf||_2$ ...
2
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1answer
23 views

How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
2
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0answers
35 views

Is the Laplacian an unbounded operator?

"The Laplacian is an unbounded operator": I read this in a book. But on Wikipedia it says: The Laplace operator $$\Delta:H^2({\mathbb R}^n)\to L^2({\mathbb R}^n) \,$$ (its domain is a Sobolev ...
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0answers
12 views

Constructing Følner sequence from invariant mean with prescribed density on a given set.

Let $G$ be a discrete countable amenable group. Suppose that $\lambda$ is an invariant mean on $G$ and $B \subset G$. Does there exists a Følner sequence $F_n \subset G$ such that $$\lambda(\chi_B) = ...
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1answer
36 views

Is $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $L^{\infty}\left(\Omega\right)$?

Let $\Omega$ be an open bounded smooth domain in $\mathbb{R}^{N}$. Let $p>N$, is $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $L^{\infty}\left(\Omega\right)$? In some textbook such as ...
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0answers
57 views

$L^p$ $L^q$ $L^{\infty}$ inclusion + Folland

Proposition 6.10 in Folland's Analysis book states: If $0 < p < q < r \leq \infty$, then $L^p \cap L^r \subset L^q$ and $\|f\|_q \leq \|f\|_p^\lambda\|f\|_r^{1 - \lambda}$, where ...
2
votes
1answer
63 views

Self-adjoint operator has non-empty spectrum.

I am trying to prove, that a self-adjoint (maybe unbounded) operator has a non-empty spectrum. So far I have argued, that if $\sigma(T)$ would be empty, $T^{-1}$ would be a bounded self-adjoint ...