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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237 views

Spectrum of the unbounded operator $i\partial_x$

I've been puzzling over this for some time now, and can't quite make my intuitions precise. I need to find the resolvent set and spectrum of the operator $$ Lu=i\frac{du}{dx} $$ taken to be ...
0
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1answer
35 views

if $Y$ and $Z$ be two closed subspace of banach space $X$ .Prove $p:X \mapsto Y$ is continuous.

Here Y and Z are closed subspace of such that $Y\cap Z=\{0\}$and $X= Y+Z$ .I have to prove that $p:X \mapsto Y$ is continuous.where $p(y+z)=y $ $\forall y\in Y$ and$\forall z\in z$. ...
3
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1answer
105 views

Give an example of a continuous linear operator $\displaystyle\|T\|=\sup_{\|x\|\le1} \|T(x)\|$ such that the supremum not reached

Let $T:X\longrightarrow Y$ be a continuous linear operator , $X \;,\;Y$ normed spaces with $$\|T\|=\sup_{\|x\|\le1} \|T(x)\|$$ Give an example of a continuous linear operator such that the supremum ...
3
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2answers
108 views

Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) ...
2
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3answers
103 views

Prove uniform convergence of $x^{\frac{1}{n}}+(1-x)^{n}$

Is it true or not that the this succession converges uniformly on $(C[0, 1],\Vert . \Vert_{\infty})$: $$f_{n}=x^{\frac{1}{n}}+ (1-x)^{n}$$I have found an elementary solution, but I would like to ...
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1answer
44 views

Is $l^{\infty}$ subpace $s(\mathbb{F})$?

Let be $s(\mathbb{F})$ a all sequence with term in $\mathbb{F}$ ($\mathbb{C}$ or $\mathbb{R}$). Denoted one element of $s(\mathbb{F})$, that $x=(x_j)$. Let be $$l^{\infty} = \{x\in s(\mathbb{F}); ...
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0answers
49 views

Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
1
vote
1answer
66 views

Prove that $X_{v+w} \subset X_v+X_w$

Let $B \subset \mathbb{R}^d$ a set convex and simetric, ($B=-B$). Prove that $X_{v+w} \subset X_v+X_w$, where $$X_v= \{ \alpha >0 \ ; \ \frac{1}{\alpha}v \in B \}$$ $$X_w=\{\varepsilon >0 \ ; \ ...
1
vote
1answer
121 views

Self-adjoint operator on a Hilbert space.

Let $T$ be a self-adjoint operator on a Hilbert space $H$. If for all $x\in H$, $\langle Tx,x\rangle=0$, is $T=0$?
10
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1answer
216 views

Proof that there is no Banach-Tarski paradox in $\Bbb R^2$ using finitely additive invariant set functions?

I am wondering if anyone is familiar with the above topic? I have found a proof that it is possible to define a finitely additive invariant set function in $\mathbb{R}^2$ on the circle in Lax's book ...
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1answer
167 views

Residual spectrum is empty

I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468) For a bounded self-adjoint linear operator ...
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0answers
39 views

Compactness in $L^p$

I am studying this article: http://arxiv.org/pdf/0906.4883.pdf There is a little part that I do not understand, in the proof of theorem 5, page 4. Let P be the projection map of $L^p(\mathbb{R}^n)$ ...
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1answer
186 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
5
votes
1answer
106 views

Zeta Regularized Determinant of Laplacian

Can anyone point me to a resource where the zeta regularized determinant of the Laplacian is explicitly computed for simple two dimensional surfaces, say a rectangle or torus or cylinder?
2
votes
1answer
76 views

Can I deal with the weak derivative in the “strong” sense?

This is an exercise in functional analysis: For $k=1,2,3$, let $A_k: D(A_k)\subset L^2([0,1])\to L^2[(0,1)]$ be the first-order differential operators $A_ku=iu'$ with domains $$ D(A_1) = ...
2
votes
1answer
77 views

About Equicontinuous and Boundedness

Let $X$ be a TVS and $X'$ denotes the space of all continuous linear functionals on $X$. Let us denote the $weak^*$-topology on $X'$ by $\sigma(X',X).$ My question is this. Why does every ...
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0answers
83 views

Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
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0answers
135 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
2
votes
1answer
226 views

Second annihilator of subspace is the subspace itself?

Let $X$ be a Banach space over $\mathbb{C}$ with dual space $X'$ and let $M,N$ be subspaces of $X',X$ respectively. Define the annihilator subspaces of $M$ and $N$ as $$ M_\circ = \{x \in X: f(x) = ...
3
votes
1answer
174 views

When is a subset of $\ell^2$ compact?

I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a ...
3
votes
1answer
164 views

Convergence of operator norm

I have a linear bounded operator $A:L_2(0,1) \rightarrow L_2(0,1)$ satisfying $\|A^n\|^{1/n} \rightarrow 0$. Thus, for some sufficiently large $N$, $\|A^N\| < 1$ and then from Gelfand's formula, I ...
2
votes
1answer
45 views

about well-defined integral kernel

Let $\phi:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ measurable function such that $$ \int_{\mathbb{R}^n}|\phi(x,y)|\ dx \leq M\ , \quad \int_{\mathbb{R}^n}|\phi(x,y)|\ dy \leq M\,.$$ Let $f\in ...
7
votes
1answer
220 views

Maximal ideals in the algebra of continuously differentiable functions on [0,1]

This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
0
votes
1answer
313 views

Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?

I wish to show the following theorem: Let $T:H\to H$ be a bounded linear operator on a complex Hilbert space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$ for all $x\in H$, then $T$ is ...
2
votes
1answer
87 views

About a Weak Topology of a Vector Space

Let $X$ be a real vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ ...
1
vote
1answer
50 views

About a Weak Topology on TVS(part 2)

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
0
votes
1answer
375 views

Operator Norm of a Linear Transformation

PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
3
votes
2answers
181 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
1
vote
2answers
140 views

Hahn-Banach theorem (second geometric form) exercise #2

Let $X$ be a Hilbert space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F),$$ and any kernel of the involved functionals is ...
2
votes
2answers
86 views

What happens when you change space of test functions associated with weak derivatives?

Recall that $u \in L^2(0,T;H^1)$ has weak derivative $u' \in L^2(0,T;H^{-1})$ iff $$\int_0^T uv' = -\int_0^T u'v$$ holds for all $v \in C_0^\infty(0,T).$ What happens if we only require that this ...
0
votes
1answer
232 views

Non-rectifiable space-filling curve

Check that the following curves γ : [0, 1] → R^2 are not rectifiable (a) γ(0) = (0,0) and γ(x) = (x,xsin(1)) for x≠0. x (b) γ is a space-filling curve: by this we mean that the image of the ...
3
votes
0answers
124 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
0
votes
1answer
132 views

How to use the Mean Value Theorem to find the “Contraction Constant”

Show that the contraction $T(x)= (1+x)^{1/3} $ on the interval $I=[1,2]$ satisfies the definition of a contraction. It's not just this problem-- on this site and others explanations will say "the ...
3
votes
1answer
262 views

Question about theorem 3.2 from Morse theory by Milnor

The demonstration of the theorem 3.2 in the book Morse theory by Milnor THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index ...
3
votes
1answer
96 views

Let $(X,\tau)$ be compact Hausdorff with $C(X,\Bbb R)$ finite dimensional. Show that $X$ is finite

If $(X,\tau)$ is a compact Hausdorff topological space so that $C(X,\mathbb{R})$ is finite dimensional real vector space, would any one help me to show $X$ is finite set? $C(X,\mathbb{R})$ denotes the ...
2
votes
1answer
92 views

norm of product of normed spaces

If $(X_1,||.||_1)$ and $(X_2,||.||_2)$ are two normed spaces and define norm on $X_1\times X_2$ as $||x||=\max(||x_1||_1,||x_2||_2)$. I want to check the triangle inequality property for this norm, ...
0
votes
1answer
182 views

prove that linear span of an orthonormal set M of a hilbert space is closed

prove that linear span of an orthonormal set M of a Hilbert space is closed I think i need a convergent seq in M and show that the limit belongs to span of M. but could not do it.
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votes
2answers
51 views

Inner product convention for $\ell^p$?

So I'm reading through some analysis problems and one is discussing $\ell^p$ (the space of $p$-summable sequences $x: \mathbb Z^+ \to \mathbb C$ such that $\sum_{n \in \mathbb Z^+}|x_n|^p < ...
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2answers
85 views

Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$

Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ (a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
3
votes
1answer
167 views

Hahn-Banach theorem (second geometric form) exercise

Let $X$ be a vector normed space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F).$$ Apply the Hahn-Banach theorem (second ...
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vote
1answer
87 views

PDEs: subsequence converges to solution, so whole sequence does too

Suppose we want existence of a function $u$ for the PDE $$(\frac{d}{dt}u,v) = b(u,v)$$ for all $v$ in a test space. Sometimes in PDE you have use a Galerkin approximation, so say $u_n$ is a sequence ...
2
votes
1answer
184 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
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vote
0answers
126 views

Properties of the derivative $\frac{\delta F(\phi(x))}{\delta\phi(y)}$ , where $ \phi(x) = F(\phi(x)) $ is a fixed point problem.

Dear experts I have a fixed point problem of the type: $ \phi(x) = F(\phi(x)) $, where $x \in \mathbb R^3 $. $\phi(x)$ is differentiable non-negative function on a given domain. I am trying to find ...
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1answer
61 views

Every almost periodic function is uniformly continuous

I know that weakly almost periodic functions an a locally compact group are uniformly continuous. But I do not know how to prove it. Would you please introduce a good reference to me? Thanks.
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1answer
79 views

Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.

$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
1
vote
3answers
537 views

Continuous Function on a Closed Bounded Set in $\mathbb{R}^n$ then that function is bounded and uniformly continuous

Theorem : Let $A$ be closed bounded set in $\mathbb{R}^n$, and let $f:A\rightarrow\mathbb{R}$ be continuous. then $f$ is bounded and uniformly continuous on $A$. I've been proved this theorem, my ...
3
votes
1answer
475 views

Examples of some linear and nonlinear operators

Let $H$ be a Hilbert space. Could you please give me examples of linear or nonlinear operators $F: H \to H$ such that $$ \limsup\limits_{\|x-y\| \to 0} \|F(x)-F(y)\| = +\infty \quad \forall x,y\in H ...
9
votes
0answers
157 views

Existence of a map in a Hilbert space

Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$. Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
3
votes
2answers
106 views

Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
1
vote
1answer
62 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...