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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3
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0answers
33 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
0
votes
0answers
50 views

If $\|f\|_X\le c_1\|f\|_Y$ then do we have $\langle f,g\rangle_X\le c_2\langle f,g\rangle_Y$?

Let $X$ and $Y$ be two inner product spaces with inner products $\langle \cdot,\cdot\rangle_X$ and $\langle \cdot,\cdot\rangle_Y$, respectively. Suppose we have $\|f\|_X\le c_1\|f\|_Y$ for any $f\in ...
0
votes
1answer
31 views

Equicontinuous sequence in $C(\mathbb{R^2})$ and Arzela-Ascoli Theorem

Could anyone help with the following problem? I am trying to work out this last practice problem for my Real Analysis prelim but I'm not sure about how to approach it. It looks very similar to the ...
0
votes
2answers
23 views

If $M$ is $F$-measurable, then is it also $F'$-measurable with $F'\subset F$?

$F$ and $F'$ are $\sigma$-algebras, and $M$ is a function from $(\Omega,F)$ to $(\mathbb{R},B(\mathbb{R}))$ If this statement is true, how to reason or understand it in a simple way?
1
vote
0answers
20 views

The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
3
votes
1answer
60 views

Gateaux derivative of $L_p$ norm

For $2\leq p < \infty$, if we consider $f,g \in L_p(X, \mathcal{M},\mu)$ there is the well-known equality $$\frac{d}{dt}\Vert f+tg \Vert_p^p = \frac{p}{2} \int_X \vert f(x)+tg(x) \vert^{p-2} ...
1
vote
1answer
30 views

The intersection of $BV$ space.

Given $\Omega\subset \mathbb R^N$ open bounded smooth boundary. We define $$ TV(u,\alpha):=\sup\left\{\int_\Omega u\operatorname{div}v\,dx,\, v\in C_c^\infty(\Omega;\,\mathbb ...
0
votes
0answers
24 views

What type of self-adjoint operator does $\hat{P}$ has to be for Green's function to result in a radial exponetial $e^{-\| x-t \|^2}$

I was reading the following paper on hyper basis function (HBF) (similar to radial basis function RBF network) and was trying to understand when is it the case that the network has radial basis ...
-1
votes
0answers
18 views

Waveles, a counter example of r-regularity. (Decreasing fast)

I was studying multi-resolution analysis when I found this counter-example that I can't check. Take the space $V_0 := \{ f \in L^2(\mathbb{R}) :supp ~\hat{f} \subset [-\pi,\pi]\}$, where the hat means ...
6
votes
2answers
110 views

Generalized convex combination over a Banach space

The Question: Is the following true? If not, what further hypotheses do I need? Let $X$ be a Banach space, and let $C \subset X$ be closed and convex. Let $P$ be a probability measure over $D$, ...
0
votes
1answer
60 views

Projections: Beppo Levi

Given a Hilbert space $\mathcal{H}$. Consider projections: $$P_\lambda\in\mathcal{B}(\mathcal{H}):\quad P_\lambda^2=P_\lambda=P_\lambda^*$$ And directed indices: ...
2
votes
1answer
30 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
3
votes
2answers
92 views

Is $L^1(X) \cap L^2(X)$ a closed subspace of $L^2(X)$ and $L^1(X)$?

Suppose that $X$ be a locally compact Hausdorff space. Could we say that $L^1(X)\cap L^2(X)$ is closed subspace of $L^1(X)$ and $L^2(X)$?
0
votes
0answers
20 views

Is $L^{\infty}(Z)$ first-countable?

Is $L^{\infty}(Z)$ first-countable? A space X is said to be first-countable if each point has a countable neighbourhood basis (local base). $L^{\infty}(Z)$ is dual of $L^{1}(Z)$ with convolution ...
-1
votes
4answers
89 views

Direct proof for convexity of $e^x$ [closed]

Is there any direct proof without using second derivative for convexity of $e^x$?
-4
votes
0answers
42 views

Two topologies are coincide [duplicate]

Let $(X,\tau)$ and $(X,\tau')$ are both metrizable topological vector space, and let $(x_{n})\subset X$ and $x\in X$ and $x_{n}\rightarrow x$ in $\tau$ topology if and only if ...
4
votes
2answers
69 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
-2
votes
3answers
57 views

Do the topologies with the same convergent sequences coincide? [closed]

Do the same sequential convergence in two topologies the result are the same topologies? Let $X$ be topological space with topologies $\tau$, $\tau'$. Let $(x_{n})\in X$ and $x\in X$ and ...
0
votes
2answers
41 views

what can you say about the solutions of the equation $y' = x^2+y^2$ just by looking at the differential equation

Can we say that the graph is symmetric about origin. Because replacing $x$, $y$ with $-x$, $-y$ does not change the equation Also the slope becomes larger as we move away from origin. Anything else ...
2
votes
0answers
29 views

Equivalent definition of uniform convexity

A Banach space $X$ is said to be uniformly convex if the following is satisfied: For $\epsilon>0, \exists \delta>0$ such that $x,y\in X, \|x\|, \|y\|\leq 1$, $\|x-y\|\geq \epsilon \Rightarrow ...
0
votes
0answers
22 views

A system is complete iff the only functional that zeros its elements is $\varphi=0$

Let $(X,\Vert\cdot\Vert)$ be a normed space. Let $\{x_n\}\subset X$. The dual space $X^\ast$ is the space of the functionals $\varphi:X\to\mathbb{R}$. Prove that $$\{x_n\}\text{ is complete in X}\iff ...
0
votes
1answer
39 views

How can I use Banach Contraction Principle to solve $Ax = b$?

Can anyone explain to me how Banach Contraction Principle (fixed point theorem) makes it easier to solve $Ax = b$?
1
vote
2answers
34 views

How can I compute the infimum of the following non linear functional

I was trying to solve this problem from previous exam of functional analysis and I am stuck Clealy $ \inf_{f \in \mathcal{M}}\phi(f) \leq 0$.(I can choose f=0) If I compute the infimum over the ...
3
votes
1answer
69 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
7
votes
0answers
53 views

Constructing an orthogonal basis with a choice of inner product [duplicate]

Given a linearly independent set of vectors in some vector space, is it always possible to construct an inner product so that the vectors are orthogonal? I know I can construct an appropriate inner ...
4
votes
2answers
101 views

Are the Hermite-Gauss functions linearly dense in $L^1(\mathbb{R})$?

The Hermite-Gauss functions ($t\mapsto H_m(t)e^{-t^2/2}$) are known to be an orthonormal basis for $L^2(\mathbb{R})$, a fortiori linearly dense in $L^2(\mathbb{R})$, and all are in the Schwartz space ...
1
vote
1answer
29 views

Sobolev norm in the definition of Sobolev spaces

I've seen the Sobolev space defined as: The Sobolev space $H^k(\Omega)$ is the set of all functions $u \in L_2(\Omega)$ for which the weak derivative $\partial^\alpha u \in L_2(\Omega)$ for all ...
3
votes
1answer
34 views

Definition of Sobolev space $H^s$ and domain of $-\Delta^s$

The spaces below are on $\partial\Omega$, the boundary of a bounded smooth domain $\Omega$. I read this in the book on page 141. Define $H^2 := \{ u \in L^2 \mid (-\Delta u) \in L^2\}$. And ...
0
votes
1answer
17 views

integral operator with degenerate kernel

Suppose I have an integral operator on $L^2$, $\int_0^1K(s,t)f(t)dt$ where K(s,t) is degenerate. Can I state that the norm of this operator equals its largest eigenvalue absolute value? As a concrete ...
2
votes
0answers
39 views

minimal distance betwen a point and and the halfspace containing a convex set

Let $L^2(I)$ be the usual $L_2$ space with $L_2$ norm and $S$ a convex and compact subset of $L^2(I)$. Suppose $g^*\notin S$ and $$\min_{f\in S} \|f-g^*\|$$ has the unique solution $f^*\in S$. ...
2
votes
1answer
42 views

Are there noncontinuous derivations $C^1(X) → ℝ$?

I’m looking for an example of a Banach space $X$ and a derivation $δ \colon C^1(X) → ℝ$ which is noncontinuous with respect to the topology of uniform convergence on $C^1(X)$, that is a $ℝ$-linear map ...
0
votes
1answer
15 views

standard mollifier (comparing the definition in Evans and wiki)

Hi I am looking at the definition of standard mollifier $\eta$ in Evans, and the $\eta$ from wiki enter link description here Have a very basic question, is the $\eta$ in Evans also compactly ...
0
votes
1answer
32 views

How can I solve the following exercise

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
1
vote
0answers
22 views

topology of uniform convergence on compacts and strong operator topology

I am trying to understand the proof for some lemma in a book, but the part the authors label as trivial is not trivial to me at all. Any help is appreciated. Here is the trivial part of the lemma: ...
4
votes
1answer
48 views

Let $f_1,f_2,\ldots, f_n$ be linear functionals on $X$. Show $f=\sum_{i=1}^n\lambda_i f_i$ iff $\bigcap \ker f_i \subset \ker f$

Problem Let $f_1,f_2,\ldots, f_n$ be linear functionals on a vector space $X$. Show that there exist constants $\lambda_1,\ldots,\lambda_n$ satisfying $$f=\sum_{i=1}^n\lambda_i f_i$$ if and only if ...
4
votes
1answer
58 views

Is $A=\{x \in \ell^2 \mid \sum_{n=1}^{\infty} \frac{x_n}{n}=0 \}$ dense in $\ell^2$

I think that the answer is no I thought quite a bit about this problem. My idea was to build a sequence $(y_n)_{n \in \mathbb{N}} \subset A$ such that given a $x \in \ell^2$ we pick the first N ...
1
vote
1answer
32 views

How can I prove Dini's theorem using the Baire Category theorem?

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 ...
2
votes
1answer
61 views

Proving that an orthonormal system close to a basis is also a basis

Let $\mathcal{H}$ be a Hilbert space and $(e_n)_{n \in \mathbb{N}} \subseteq\mathcal{H}$ be an orthonormal basis and $f_n$ be an orthonormal system such that $(f_n)_{n \in \mathbb{N}} ...
1
vote
1answer
78 views

Understanding a proposition in Zeidler's book on functional analysis [closed]

In my book on functional analysis by Zeidler he makes an important claim that Let $M \subset \mathbb{X}$, where $\mathbb{X}$ is a normed space then $M$ is closed $\forall u_n \in M, ...
3
votes
1answer
69 views

The norm of linear functional $x\mapsto \sum_{n=1}^{\infty} \frac{x_n}{2^n}$ on $c_0$

Consider the mapping $\phi :c_0 \to \mathbb{R}$ defined by $\sum_{n=1}^{\infty} \frac{x_n}{2^n}$. Compute $\|\phi\|$ Does there exist a $x \in c_0$ such that $\|x\|=1$ and $\|\phi\|=|\phi(x)|$ ...
4
votes
0answers
30 views

A question about equivalence of norms involving infimum

Let $I$ be a Banach space with norm $\lVert\cdot\rVert_I$. The norm $$\inf\{\lVert(G_i(u_i))_i\rVert_{\ell^2}\mid u=\sum_{I \geq 0}u_i\}\qquad\text{is equivalent to}\qquad \lVert{u}\rVert_{I}$$ where ...
5
votes
2answers
64 views

Is an $L^p$ function in an annulus $L^p$ restricted to almost all planes?

Let $n\geq3$ and consider the annulus-like domain $A=B(0,1)\setminus B(0,r)\subset\mathbb R^n$. Take any number $p\in[1,\infty]$. If $f\in L^p(A)$, is it true that $f|_{P\cap A}\in L^p(P\cap A)$ for ...
0
votes
2answers
20 views

What is $H_0^1$ space?

I'm reading a book and it says that the $H_0^1(\Omega)$ space is defined as "the completion of $C_0^\infty(\Omega)$ w.r.t the Sobolev norm $|| \cdot ||_1$, where $C_0^\infty(\Omega)$ is the space of ...
1
vote
0answers
20 views

Is $\{{\operatorname{sinc}}\big({z}-n\big)\}_{n\in \mathbb Z}$ a Riesz basis for $PW_\pi$?

N.K. Bari has proved (see N.K. Bari, "Biorthogonal systems and bases in Hilbert space" Uchen. Zap. Moskov. Gos. Univ. , $148$ : $4$ ($1951$) pp. $69–107$) that a system $\{\phi_n\}$ is a Riesz system ...
-1
votes
2answers
36 views

Difference quotients and weak derivatives (Evans 5.8.2 theorem 3) [closed]

Could anyone give a proof on the remark below the theorem? Basically it is problem 11 I think the proof relates to example 19.19 (p.375) in enter link description here But I really do not ...
4
votes
0answers
33 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
0
votes
1answer
26 views

Spectral Measures: Pushforward

This thread is Q&A. Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
1
vote
1answer
29 views

Continuity of multiplication of operators in the strong operator topology - find an error

I need help in finding the mistake in the following reasoning. I proved that if dimension of Banach space $X$ is infinite, then multiplication of bounded operators is separately continuous but not ...
1
vote
1answer
20 views

Bound on the product of functions in $L^1$

Let X be a bounded subset of $\mathbb{R}$ and let $f, g,$ and $h$ be real valued functions in $L^2(X)$. Consider $$\| fgh\|_{L^1(X)}.$$ The hope is to get an upper bound in terms of ...
2
votes
2answers
38 views

How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...