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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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537 views

Spectrum of the operator

Let $T$ be an operator on Hilbert space. Define $\sigma(T)=\lbrace \lambda\in \mathbb{C} | \lambda I - T~\textrm{is not invertible}\rbrace$. How can I prove that $\sigma(T^n)=\lbrace \lambda^n|\...
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vote
1answer
1k views

Ky Fan Norm Question

How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix) Thanks.
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vote
1answer
203 views

Implicit functions and Quotient Rule

In "Introduction to Calculus and Analysis" pages 221-223 Courant derives the following for an implicit function F(x,y)=0. Using $dF = F_x dx + F_y dy = 0$ $dy = \frac{dy}{dx} dx = -\frac{F_x}{F_y}...
7
votes
2answers
1k views

Topologies on the space $\mathcal D'(U)$ of distributions

In my analysis lecture I am given a topology on the space of distributions as follows: Let $u_k$ be a sequence in $\mathcal D'(u)$, $u \in \mathcal D'(u)$. We say $u_k \rightarrow u$, if $\...
13
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3answers
3k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
0
votes
1answer
290 views

Image Of a Discontinuous linear functional

I want to show, Image of $unit~ disc$ under a discontinuous linear functional from a Normed Linear Space is $\mathbb{C}$ . I know the Image is Unbounded
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votes
3answers
356 views

Is a sequence of disjointly supported functions in $L^\infty$ complemented?

Let $(f_n)_{n \geq 1}$ be disjointly supported sequence of functions in $L^\infty(0,1)$. Is the space $\overline{\mathrm{span}(f_n)}$ (the closure of linear span) complemented in $L^\infty(0,1)$? By ...
11
votes
2answers
6k views

Compactness in the weak* topology

Let $X$ be a Banach space, and let $X^*$ denote its continuous dual space. Under the weak* topology, do compactness and sequential compactness coincide? That is, is a subset of $X^*$ weakly* ...
7
votes
1answer
501 views

Isometries of $\ell^p_n(\mathbb{C})$

Let $1<p<\infty$, and define an isometry of normed linear spaces to be a norm-preserving surjection. Then all isometries from $\ell^p_n(\mathbb{R})$ to itself are given by linear transformations ...
4
votes
1answer
330 views

Simultaneous orthogonal basis for $L^2$, $H^1_0$, … $H^k_0$

Let $\Omega \in R^n$ be open, bounded and with smooth boundary. Can you prove the existence of a system of vectors that simultaneously forms an orthogonal basis both in $L^2(\Omega)$ and $H^1_0(\Omega)...
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3answers
168 views

How can I write $\frac{1}{(a+x)}$ as an exponential function $y = Ce^{-kx}$?

How can I write $\frac{1}{a+x}$, $a$ a non-zero positive constant, in exponential terms in the form of $y = Ce^{-kx}$? I've tried to use to Taylor series but they only seem to work for $x < 1$.
2
votes
1answer
213 views

Extrapolation of an analytical function

Given a function $f:z\mapsto f(z)$ for a discrete set of points in the real interval $z\in[a,b]$ and the knowledge that $f$ is analytical along the real axis and that its Fourier transform is real ...
3
votes
2answers
593 views

On the properties of the Sobolev Spaces $H^s$

Let $H^s(\mathbb{R}^d):= \{ u \in \mathcal{S}' : (1+|\xi|^2)^{s/2}\hat{u}(\xi) \in L^2(\mathbb{R}^d)\}$. It can be shown that this space is a Hilbert space and that $H^s \subset H^t$ if $t \leq s$....
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votes
2answers
16k views

When is the image of a linear operator closed?

Let $X$, $Y$ be Banach spaces. Let $T:X\rightarrow Y$ be a bounded linear operator. Under what circumstances is the image of $T$ closed in $Y$ (except finite-dimensional image). In particular, I ...
5
votes
1answer
526 views

Bounding the integral of a $C^1$ function using its gradient

Let $f \in C^1_c(\Omega)$ where $\Omega \subset \mathbb{R}^d$ is a bounded domain. Let $\phi \in C^1_c(\mathbb{R}^d)$ be an approximation of the identity (i.e. $\int_{\mathbb{R}^d} \phi=1$, $\phi \geq ...
4
votes
1answer
779 views

How to convert a powerseries into a Dirichlet series?

[Update 3]: I realized, that the series $s(h)$ below is simply a "(general) Dirichlet-series" (1), so after I know the principle how to find the Taylor-series for some function (which may be given by ...
5
votes
1answer
918 views

Measure Theory and Integrals of Characteristic Functions

Given two sets of finite measure in $\mathbb{R}$ say, $E$ and $F$, and their characteristic functions $\chi_E$ and $\chi_F$, can somebody show that $\chi_E\ast\chi_F(x)$ (the convolution) is a ...
5
votes
3answers
2k views

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard ...
2
votes
2answers
2k views

Hilbert spaces, square integrability etc

(Someone may please change the title if they can think of a better one) We have a Hilbert Space $\mathcal{H}$ that consists of all functions $\psi(x)$ such that $\int_{-\infty}^{\infty} |\psi(x)|^2 ...
8
votes
1answer
376 views

What is the ''right" norm for the Banach space tensor product in this situation?

Let $X,Y$ denote (real) vector spaces. The vector space of $n$-linear maps $X^n \to Y$ will be denoted by $L^n(X,Y)$. Unless I'm much mistaken $$L(X,L(X,Y)) \ \ \ L^2(X,Y) \ \ \ L(X \otimes X,Y)$$ ...
4
votes
1answer
199 views

Hardy Spaces on the unit disk and $\mathbb R^n$

Is there a connection between the Hardy Spaces on the unit disk and on $\mathbb R^n$?. If so, can we use results from the Hardy Spaces on the unit disk to prove $(H^1)^* = \text{BMO}$? Further, what ...
5
votes
2answers
736 views

construction of a linear functional in $\mathcal{C}([0,1])$

Can someone help me to construct a linear functional in $\mathcal{C}([0,1])$ that does not attain its norm? Actually, I want to prove that $\mathcal{C}([0,1])$ is not reflexive Banach space. Is it ...
14
votes
2answers
10k views

Closed Subspaces of Vector Spaces

Question: In Functional Analysis we can note things like: every closed subspace of a Banach space is Banach. In this case, what does "closed subspace" mean? Does this mean closed under the norm ...
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vote
1answer
184 views

Euler-Lagrange expression uniformly non-negative

If you form the Euler-Lagrange equation for some calculus of variations problem in x, f(x) and f'(x), and the resulting expression is always non-negative over the domain of x (because the expression ...
5
votes
2answers
4k views

Weak convergence in Sobolev spaces

Consider the inner product by $\langle f,g \rangle_{H^1} = \langle f, g \rangle_{L^2} + \sum_{|\alpha|=1} \langle D^\alpha f, D^\alpha g \rangle_{L^2}$ where $\alpha$ is a multi-index and $D$ denotes ...
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2answers
3k views

On the limits of weakly convergent subsequences

Let $\{ f_n \}$ be a sequence in a Hilbert space $L^2(\mathbb{R}^d)$. We say that this sequence converges weakly to an element $f \in L^2$ if $\langle f_n, g \rangle \to \langle f,g \rangle$ for every ...
4
votes
2answers
746 views

Hilbert Space - Norm of derivative

If $H$ is a Hilbert space of entire functions with weighted norm $||f||^{2}=\int_{R} |\frac{f(t)}{g(t)}|^{2}dt$ for some entire function $g$ (not necessary in $H$). Can we find any relation between ...
8
votes
1answer
5k views

How does the method of Lagrange multipliers fail (in classical field theories with local constraints)?

The method of Lagrange multipliers is used to find the extrema of $f(x)$ subject to the constraints $\vec g(x)=0$, where $x=(x_1,\dots,x_n)$ and $\vec g=(g_1,\dots,g_m)$ for $m \leq n$. Although ...
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2answers
2k views

Some basics of Sobolev spaces

Let $W^{m,p}(\Omega) = \{ f \in L^p(\Omega): D^\alpha f \in L^p(\Omega) \text{ for multi-indices } |\alpha| \leq m\}$, where $D$ denotes the weak derivative. Let $W_0^{m,p}$ denote the closure of $C_c^...
1
vote
1answer
337 views

Eigenvalues of a product of similar operators

The operators in this question are in Hilbert space, but I'm going to word it as a discrete spectrum and a finite dimensional (linear algebra) answer is fine by me. Operators $A$ and $\Delta$ are ...
3
votes
2answers
172 views

How to prove that the sequence of $x_n = (1,\frac{1}{2}, \frac{1}{3}, … \frac{1}{n}, 0, 0…)$ does not converge under $\|\cdot\|_1$?

I'm reviewing past assignments and am still having trouble formulating a proof for this: Consider the sequence $(x_n)$, where $x_n = (1,\frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, 0, 0, \ldots)$. ...
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2answers
1k views

On the density of $C[0,1]$ in the space $L^{\infty}[0,1]$

It's easy to show $C[0,1]$ is not dense in $L^{\infty}[0,1]$ in the norm topology, but $C[0,1]$ is dense in $L^{\infty}[0,1]$ in the weak*-topology when take $L^{\infty}$ as the dual of $L^{1}$. how ...
7
votes
2answers
643 views

On the isometry between bounded linear operators and the dual of nuclear linear operators

Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible $$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i \...
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votes
1answer
312 views

Form of the inner product in $\ l_2$

Is it true that every inner product in $\ l_2 $ is of the form $\langle x,y\rangle_a =\sum_{n=1} ^ {\infty} {a_n x_n y_n}$ ? (Of course $\ x=(x_n) , y=(y_n) $ are in $\ l_2 $ .)
5
votes
1answer
189 views

Ergodic Recurrence

My solution concerning a problem about Ergodic Recurrence requires me to prove that $\|P_T 1_B\| > 0$. Where $P_T$ is the projection onto the space $I := \{f \in L^2 : f \circ T = f\}$, $T$ is a ...
9
votes
1answer
2k views

Why are inner products in RKHS linear evaluation functionals?

I'd like to know why inner products in Reproducing kernel Hilbert spaces are (linear) evaluation functionals. I understand that inner products are linear functionals, and I know what an evaluation ...
4
votes
1answer
699 views

Preservation of Lipschitz Constant by Convolutions

The following is a step in a proof: $f$ is a Lipschitz function from $E$ to $F$ where $E$ is a finite-dimensional Banach space and $F$ an arbitrary Banach space. $\phi\geq 0$ is a $C^\infty$ function ...
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1answer
2k views

Nested sequences of balls in a Banach space

This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help. (By the way- this question does come from home-work, but I've ...
16
votes
1answer
672 views

What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces. Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$. I think this induces a fiber ...
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vote
1answer
129 views

Eigenvalues of a Parametrized Family of Linear Functions

Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number. For each $\alpha$, it is given that $L(\alpha)$ is a ...
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votes
2answers
467 views

Does strong convergence to 0 implies convergence to zero on compact sets?

I've been struggling with this for a bit and was wondering if anyone can give me a hint: Suppose $\{ T_n\}_1^\infty\subset \mathcal{L}\{ X,Y\}$ is a sequence of bounded operators from a banach space $...
5
votes
1answer
677 views

Uniform mean ergodic theorem

I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following $$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$ ...
171
votes
4answers
42k views

Norms Induced by Inner Products and the Parallelogram Law

Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
4
votes
1answer
981 views

$\omega$ - space of all sequences with Fréchet metric

I'm working on to prove the following: Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence. Any hint is ...
9
votes
2answers
4k views

On Some Properties of Hölder Continuous Functions

The function space $H^{\alpha} (\Omega)$ for $0 < \alpha \le 1$, is the set of functions: $$\{ f \in C^0(\Omega) : \sup_{x \neq y} \dfrac{|f(x) - f(y)|}{|x-y|^{\alpha}} < \infty \}$$ with the ...
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vote
0answers
239 views

Rotund and smooth space

Suppose $X$ is normed space. Prove that $X^\ast$ is rotund iff $X/M$ smooth whenever $M$ is a closed subspace of $X$ such that $X/M$ is two-dimensional. I know that "a normed space is smooth iff each ...
4
votes
2answers
242 views

Uniform Boundedness/Hahn-Banach Question

Let $X=C(S)$ where $S$ is compact. Suppose $T\subset S$ is a closed subset such that for every $g\in C(T),$ there is an $f\in C(S)$ such that: $f\mid_{T}=g$. Show that there exists a constant $C>0$ ...
4
votes
1answer
308 views

Matrix Analysis in $\mathbb{C}$

1.Let $A \in GL_n(\mathbb{C})$. Show that $\det(I+A)=1+\operatorname{tr}(A)+ \epsilon(A)$ where Modulas of epsilon(A) by norm of A=0 as A tends to 0,for any matrix norm. If I define J(A)= det(A) for A ...
10
votes
1answer
859 views

Spectrum of composition of convolution and multiplication

Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ defined as composition of convolution and multiplication, $Tf := (af) * g$ where $g$ is in $L^2$ and $a$ is a bounded function. ...
21
votes
2answers
2k views

When is a notion of convergence induced by a topology?

I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...