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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How to show $e^{-x}$ is a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ in $L^{2}[0,\infty)$?

Let $\mathcal{H}=L^{2}[0,\infty)$. How can one easily show that $e^{-x}$ is a cyclic vector under the $C^{\star}$ subalgebra of operators on $\mathcal{L}(H)$ generated by all resolvents $(L-\lambda ...
2
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2answers
38 views

Proofs that rely on an infinite matrix

If I have an operator $A\in B(\mathcal{H})$ that can be "identified" with an infinite matrix with countably many entries, is it in any way unrigorous to do actual calculations with the picture we have ...
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2answers
48 views

Is $\ell^1$ an inner product space?

Considering the parallel result for $L^p$ spaces, I would guess that $\ell^1$ is not an inner product space. The proof would presumably follow by providing counterexample sequences $x = (x_n) $ and ...
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22 views

Non-negative supremum implies existence of functional.

$\ell_\infty(T) = \{f:T\to\mathbb{R} : \sup_{t \in T}|f(t)| < \infty\}$. Let $G$ be a family of transformation of set $T$ and $V$ linear subspace of $\ell_\infty(T)$ such that: $1_T \in V$, ...
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1answer
24 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $\mathcal{L}^2(\mathbb{R})\cap\mathcal{L}^1(\mathbb{R})$ and thus we can compute the Fourier transform $$\hat{f} (\xi) = ...
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12 views

Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form T_0 = i\alpha grad + \beta, where \alpha and \beta are Hermitian 4 x 4 matrices, and T_0 is selfadjoint on some domain ...
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2answers
28 views

Conditions for a given operator being compact.

So I was given this question in class, and I thought it looked easy enough at first glance, but actually trying to do it, I have gotten quite stuck on the "only if" part. Let $T\colon \ell_\infty ...
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22 views

Which projection, in $L_\infty$ norm or $L_2$ norm, is non-expansion?

I am just wondering which projection is non-expansion? Is there any reference to find this out.
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17 views

Equivilent expression of spectral measure

Let $E_{\lambda}$ be any spectral family associated with the semibounded selfadjoint operator $H$ such that $$ H=\int_{0}^{\infty}{\lambda dE_{\lambda}} $$ If we denote $T_{z}(x)=\frac{\lambda ...
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0answers
19 views

Mysterious Commutation in an Unbounded Operators Argument (Compact Resolvent) — Is there a typo?

While reading A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Chapter 1, Section 4, I came across a passage that I really do not understand, and I am trying to see if ...
2
votes
1answer
45 views

A linear operator is continuous if and only if it maps cauchy sequences to cauchy sequences

Let $A$ and $B$ be seminormed spaces, then I want to show that a linear operator $T: A \rightarrow B$ is continuous if and only if it maps cauchy sequences to cauchy sequences. The direction "$T$ ...
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1answer
21 views

Trace of $L^p$ function

For $U$ a bounded domain in $\mathbb{R}^n$, why is it that, in general, an $L^p$, $1\leq p<+\infty$, function does not have a trace on the boundary of $U$? Thanks in advance.
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17 views

functional analysis fix point

what provide Tychonoff fixed point theorem (let $C$ be a nonempty compact convex subset of a locally convex topological linear space $X$ and $T: C\rightarrow C$ a continuous mapping , then $T$ has a ...
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0answers
19 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
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1answer
10 views

Riesz map on $L^2(0,T;H)$ — it's not “unique” in a way

Let $R:L^2(0,T;H) \to L^2(0,T;H^*)$ denote the Riesz representation map. Given $u \in L^2(0,T;H)$, $Ru \in L^2(0,T;H^*)$ can be changed in $[0,T]$ on a set of measure zero. So $Ru$ is not unique in ...
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3answers
26 views

extending a bounded linear operator

So I have a homework question which I have no idea how to start. Let $E_0$ be a dense subspace of the normed space $E$. Let $T_0:E_0 \rightarrow F$ be a bounded linear operator into the Banach space ...
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2answers
45 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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30 views

Proof: $f$ square-integrable $\Rightarrow f$ absolutely integrable on $[0, 2\pi]$

In a book I found the following statement: Let $\varphi(x)$ and $\psi(x)$ be square integrable, then $|\varphi \psi| \leq \frac{1}{2} |\varphi^2 + \psi^2|$. This implies, that every square ...
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3answers
44 views

Why $L^1$ is not reflexive [duplicate]

We already known that $$ (L^p(\Omega))^* = L^q(\Omega), $$ for all $1\le p < \infty $ and $q$ is the exponent conjugate to $p$. So that, $L^p(\Omega)$ is reflexive with $1<p<\infty$. However, ...
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2answers
18 views

How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
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0answers
17 views

Given that the support function $\sigma_S(x)$ is finite-valued, prove that $S$ must be bounded (without the use of Uniform Boundedness Principle)

Here is the problem: Let $S\subset \mathbb{R}^n$ and consider the support function $\sigma_S:\mathbb{R}^n\to\mathbb{R}^n\cup\{\pm\infty\}$ defined by $$\sigma_S(x)=\sup_{s\in S}(sx)$$ Prove, ...
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0answers
11 views

Bump Functions on Open Intervals

I just have a quick question about bump functions. If we're dealing with $C^{\infty}_{0}((0,\infty))$, i.e. all smooth bump functions on $(0,\infty)$, obviously any $f\in C^{\infty}_{0}((0,\infty))$ ...
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0answers
16 views

Separable $X^*$ Property

Let $X$ be a normed space. If $X^*$ is separable, then there exists $(f_n)_{n\geq1}\subset X^*$ such that $\|f_n\|_{X^*}=1$ for all $n$ and $\{f_n:n\geq1\}$ is dense in $\{f\in X^*:\|f\|=1\}$. In ...
3
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0answers
30 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
4
votes
0answers
34 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
0
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1answer
13 views

Problem expressing that because a given linear functional is unbounded, that a norm inequality cannot hold.

Here is my problem: Let $X$ be a Banach space with norm $\|x\|$. Let $\phi:X\to\mathbb{R}$ be a non-zero, unbounded linear functional. Prove that $\|x\|_\phi :=\|x\|+|\phi(x)|$ defines a different ...
0
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2answers
31 views

$X$ is a normed vector space and $T:X\to X$ is a function that has a closed graph, does $T$ map closed sets to closed sets?

Here is the question: Suppose that $X$ is a normed vector space and $T:X\to X$ is a function that has a closed graph. Is it true that $T$ maps closed sets to closed sets? Is it true if $T$ is linear? ...
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1answer
23 views

Given $X,Y$ are Banach spaces with norms $\|x\|_X,\|y\|_Y$, prove $\|(x,y)\|=\max(\|x\|_X,\|y\|_Y)$ is a norm and defines a Banach space

Here is the question: Let $X$ and $Y$ be Banach spaces with norms $\|x\|_X$ and $\|y\|_Y$ respectively. Prove that $$\|(x,y)\|=\max\{\|x\|_X,\|y\|_Y\}$$ defines a norm on $X\times Y$, and that ...
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20 views

Isometric isomorphisms between normed spaces and compact hausdorff spaces

Let $X$ be a normed space. Show that there is a compact Hausdorff space $Y$ such that $X$ is isometrically isomorphic to a subspace of $C(Y)$. I think this might be proved using the Banach–Alaoglu ...
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1answer
19 views

Embedding of Lp spaces

I've managed to prove that for $ 1\leq p < q \leq +\infty $ we have an inclusion (embedding) $ L_q([0,1],\lambda) \rightarrow L_p([0,1], \lambda) ~~ (\lambda $ being Lebesgue measure). The trouble ...
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1answer
21 views

Discontinuous mapping between function spaces

Let $ C([0,1]) $ be a space of continuous real-valued functions over interval $[0,1]$ and $ \|f\|_2 = (\int_0^1|f|^2 \, dx)^{1/2} $ define a norm over this space. Prove that the following mapping: $ ...
0
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1answer
20 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
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1answer
12 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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1answer
23 views

Proving an orthonormal set is an orthonormal basis in Hilbert space [duplicate]

Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also ...
0
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1answer
9 views

Showing T intertwines $D_T$ and $D_{T^*}$ using Spectral Theorem

Suppose $T$ is a contraction on a Hilbert space $H$ (separable, if you wish). $D_T=(I-T^*T)^{1/2}$ and $D_{T^*}=(I-TT^*)^{1/2}$. I want to show that $TD_T=D_{T^*}T$. I had done this before using a ...
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0answers
21 views

Compositions and products on Sobolev spaces

Does anybody have a good textbook reference for someone who wants to begin studying products and compositions in Sobolev spaces, where the underlying domain is either $\mathbb{R}^n$ or an open subset ...
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11 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
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1answer
57 views

A lemma by Foguel and Weiss [1973]

so I am reading Krengel's text on Ergodic theorems. And the next lemma bugs me as for the proof of it. It's by Foguel and Weiss. Statement: If $P_1, P_2$ are commuting elements of a Banach algebra ...
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30 views

A complicated question

I have the following operator $A: H^1_{0,p}\longrightarrow H^1_{0,p}$ be defined by \begin{equation} Au(t)=\int_0^{+\infty} G(t,s)q(s)f(s,u(s))\,ds-\sum_{k=0}^{+\infty}G(t,t_k)h(t_k)I(u(t_k)), ...
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1answer
60 views

T is not compact operator

I want to show that if $T$ is a bounded operator between two Hilbert spaces and $T$ is not compact then there exists an orthonormal sequence $y_{n}$ and an $R>0$ such that $\forall n\in ...
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26 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
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1answer
16 views

Help with the proof of the open mapping theorem

I don't understand the following from the proof of the open mapping theorem. Suppose $A $ is a bounded linear transformation from the Banach space $X $ onto the Banach space $Y $. Using Baire's ...
2
votes
1answer
48 views

$p$-summable series in a Banach space

Let $E$ be a Banach space and denote its dual space by $E^*$. Let $p \in [1, \infty)$ and $x : \mathbb{N}\rightarrow E$ be such that for every $\phi \in E^*$, $$\left( \sum_{n=1}^{\infty} \lvert ...
2
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1answer
27 views

Are the normed spaces $ \mathbb{R}^{n^2} $ and $ M_n(\mathbb{R}) $ isometric?

Consider the spaces $ \mathbb{R}^{n^2} $ with euclidean norm and $ M_n(\mathbb{R}) $ of $n\times n$ matrices with the norm defined by $ \Vert A\Vert = \sup\limits_{\Vert x\Vert \le 1}\Vert Ax\Vert$. ...
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1answer
23 views

$L^p$ norm of a gradient

Suppose $f:\mathbb{R}^n\to \mathbb{R}$ and let $Df=(f_{x_1},f_{x_2},..., f_{x_n})$, the gradient of $f$. A special case of the Gagliardo-Nirenberg inequality says that $$||f||_{p^*}\leq ...
0
votes
1answer
37 views

is it true that $\int\limits_{x_{n}}^xf(z)dz\longrightarrow 0$

If $x_{n}\longrightarrow x$ then is it true that $\int\limits_{x_{n}}^xf(z)\,\mathrm dz\longrightarrow 0$? We have that $f\in L_{2}(0,\infty)$ and takes complex values. I think that it is, but why? In ...
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0answers
21 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
2
votes
1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
2
votes
1answer
16 views

One Note about One to one and Surjective of linear functional [on hold]

I read a note that: if $ f \neq 0$ is a linear functional on H, then f is onto (surjective) and it is not one to one (injective) in general. Why this is true? i think it need advance ...
2
votes
0answers
22 views

Equivalent definitions of the trace of a Hilbert-Schmidt operator

I am currently reading the book Spectral Methods in Automorphic Forms, and Iwaniec defines the trace operator in a different way than I am accustomed to. Throughout, assume that everything converges ...