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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11
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1answer
150 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
3
votes
2answers
97 views

Justifying that $S=\left\{f \in X: \int f(t)dt=0\right\}$ is compact and connected

Consider the space $X=C[0,1]$ with its usual sup-norm topology. Let $$S=\left\{f \in X: \int f(t)dt=0\right\}.$$ Justify: S is compact. S is connected.
1
vote
1answer
87 views

$\ell^p(I)$ space and a dense set of this space

Let $I$ be an infinite set and $1\leq p<\infty$. Show that $\ell^p(I)$ has a dense set of the same cardinality as $I$. For this I put $$X=\{(x_i); x_i\in \Bbb C , x_i=0 \text{ for all but ...
1
vote
1answer
75 views

Boundedness and compactness of a subset of $( \ell^2 , \|\cdot\|_2 )$

Investigate the boundedness and compactness of the following subset of $( \ell^2 , \|\cdot\|_2 )$: $$S:=\{x=(x_n) \in \ell^2 : | x_n | \le 1/\sqrt{n} \ \forall n\in N\}.$$ Can the following be ...
1
vote
2answers
74 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
0
votes
1answer
52 views

Fourier series using other bases?

The theory of Fourier series, representing a reasonable function by an infinite sum of exponential functions, is very well-developed. In addition to basic functional-analytic results there are things ...
6
votes
1answer
305 views

Is Reflexivity Necessary for the Weak and Weak* Topologies to Coincide?

Let $X$ be a normed vector space, not necessarily Banach. Suppose that $X$ is not reflexive, implying the existence of such $\varphi\in X^{**}$ ($X^{**}$ being the double dual of $X$) of that for any ...
1
vote
1answer
66 views

Given an arbitrary sequence {$x_n$} in $\Bbb{R}$, find a test function $f$ with $f^{(n)}(0)=x_n$

Given an arbitrary sequence {$x_n$} in $\Bbb{R}$, can I find a test function having the $n$-derivative equal to $x_n$ at $0$?
0
votes
1answer
88 views

A basic question about function space

Im reading Carothers' Real Analysis, 1ed. Actually, Carothers has begun his talk of function space since chapter9. However, I haven't found any definition about function space. Here is a definition ...
1
vote
1answer
164 views

Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
7
votes
2answers
244 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
1
vote
1answer
192 views

The Trace Class Operators Form a Banach Space

I want examining the trace class operators $L_1(H)$ of a separable Hilbert space $H$ with the norm $||A||_1=\sum\limits^{\infty}_{n=1}\lambda_n$ where $\lambda_n$ are the eigenvalues of ...
1
vote
1answer
140 views

Application of Brouwer fixed point theorem, why is compactness not required here??

Define a map $K:X_n \to X_n$ where $X=\text{span}(v_1, ..., v_n)$ where $v_i$ are basis functions some Hilbert space $H$. So $X_n$ is finite-dimensional. $B_r(0)$ denotes the ball of radius $r$ ...
2
votes
2answers
133 views

non zero linear functional and which of the following statements are true. (NBHM-$2014$)

Let $V$ be finite dimensional real vector space and let $f$ and $g$ be non zero linear functionals on $V$. Assume that $\ker(f) \subset \ker(g).$ Which of the following are true?? a. ...
2
votes
1answer
63 views

Show that E\H (Hyperplane) is arc-connected $\Longleftrightarrow$ H isn't a closed subspace

Good morning, Let $E$ be a real normed vector space and $H$ a hyperplane of $E$ Show that E\H is arc-connected $\Longleftrightarrow$ H isn't a closed subspace I have no idea to solve it. But If $f$ ...
3
votes
2answers
252 views

Linear operators with no adjoint

Here is a standard theorem about bounded operators: Let $H$ be a Hilbert space. For any bounded linear operator $A:H\to H$ there is a unique bounded operator $A^*$ s.t $\langle Au,v\rangle=\langle ...
0
votes
1answer
74 views

Gateaux differentiable

Let $E$ is a subset of $\mathbb{R}^n$ and Lebesgue measurable respect to $\mu$ is the Lebesgue measure in $\mathbb{R}^n$, for any $p\in [1,\infty)$, let $$ f(u) = \int_E |u|^p\;d\mu\qquad \;\forall\; ...
5
votes
3answers
161 views

Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
0
votes
0answers
24 views

Inner product spaces, examples for subspaces with certain properties [duplicate]

Let $H$ be a inner product space. Give examples for a subset $U\subset H$ so that (a) $\overline{U}\neq U^{\bot\bot}$ (b) $\overline{U}\oplus U^{\bot}\neq H$ I have thought about $H=C[0,1]$ with ...
3
votes
2answers
108 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
1
vote
1answer
71 views

examples of Matrix norms induced by inner products

I'm learning about Hilbert spaces and found that, in the case of matrices as elements of the space, the Frobenius inner product induce the Frobenius norm. I'm interested in knowing more examples of ...
9
votes
1answer
1k views

Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble ...
0
votes
1answer
153 views

Frechet Derivative: Why bounded (linear) operator?

Why do we want the frechet derivative to be a bounded linear operator? (This meant more as a collecting ideas - I know bounded operators behave fine but that would exclude alot of examples such as the ...
0
votes
1answer
27 views

Pitt's theorem on automatic compactness of bounded operators between sequence spaces

Why is it called Pitt's theorem? I couldn't locate the origin of the statement.
2
votes
1answer
50 views

Is $L^\infty(0,T;V)$ a reflexive space? Question about weak convergence

Let $V$ an Hilbert space and $T>0$. Is $L^\infty(0,T;V):=\{v:[0,T]\to V: \text{ess}\,\text{sup}_{t\in [0,T]}||u(t)||<\infty$ a reflexive space? I think that since the $L^\infty$ isn't ...
1
vote
2answers
61 views

Compactness of the solution operator

Let $\Omega$ be a smooth open bounded subset of $\mathbb{R}^n$. The bilinear form $$a(u,v)=\int_{\Omega}\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}dx$$ is elliptic on ...
5
votes
2answers
174 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
15
votes
5answers
578 views

Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only?

Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only and not for $p$ any real number?
5
votes
3answers
216 views

The equation $\,\,\Delta u+\cos u=0\,\,$ possesses a weak solution in $\,W^{1,2}_0(D)$

Let $D$ be an open bounded subset in $\mathbb{R}^{n}$, with sufficiently smooth boundary. Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $$\Delta u+\cos u=0.$$ Help me ...
1
vote
1answer
131 views

Differential equations, integral equations

Is there an analytical way of proving that if $\phi$ is a solution to \begin{equation} y(t)=e^{it}+a\int_{t}^{\infty}\sin (t-s)y(s)s^{-2}ds, \end{equation} then $\phi$ would be a solution to the ...
1
vote
1answer
54 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
1
vote
1answer
84 views

Confused about class notes on gradient inequality and how to derive version for functions with compact support

I'm taking a grad analysis class, and I'm a little confused about how to show the basic gradient inequality for smooth functions with compact support. Here is my professor's statement of the basic ...
0
votes
1answer
148 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
2
votes
0answers
74 views

Space of Continuous mappings to metric spaces

I want to ask whether some basic result from the space $C([0,1],R)$, where $R$ is the real space carries over to the space $C([0,1],E)$, where $(E,\|\cdot\|_E)$ is a metric space. We know that ...
19
votes
2answers
519 views

In $\ell^p$, if an operator commutes with left shift, it is continuous?

Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution. What we have: ...
4
votes
0answers
202 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
2
votes
1answer
183 views

Orthogonal complement examples

I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that (i) $\bar{U} \oplus U^\perp \neq H$ (ii) $ \bar{U} \neq U^{\perp \perp}$ Since finite and closed ...
1
vote
1answer
100 views

Continuity of integral w.r.t. Lebesgue measure

Let $f:\Bbb R^n\times \Bbb R^m\to \Bbb R$ be a bounded measurable function, and $\mu$ be a probability measure on $\Bbb R^m$ which is absolutely continuous w.r.t. Lebesgue measure $\lambda$, e.g. $m ...
2
votes
1answer
65 views

A question on operator theory

Let $T$ be a quasinilpotent operator acting on a separable Hilbert space $H$. Fix a vector $x$ in $H$ such that $[T^n x]=H$ (the closed span of the orbit is $H$), and a hyperplane $Z\subset H$. Can we ...
2
votes
0answers
67 views

When is a function space a Fréchet space?

Let $Q$ be a space of indices, and let $(V, |\cdot|)$ be a Banach space of values. Define the function space $X = C(Q,V)$, and equip it with the topology generated by seminorms $\|x\|_D := \sup_{d \in ...
1
vote
0answers
66 views

What is the correct notation for defining norms in measure spaces?

For a function $f \in L_2(R)$, we can define its norm as $$ \|f\|_2^2 = \int f^2(x) dx $$ If I use a different measure $\mu$, I can in turn define the norm as $$ \|f\|_2^2 = \int f^2(x) \mu (x) dx ...
2
votes
1answer
27 views

An operator with infinite deficiency index

I'm looking for a simple example of an operator with infinite deficiency index .
1
vote
0answers
61 views

Weak Compact and separable sets

Is true the following statement? Se $(X\|\cdot\|)$ a Banach espace, and $K\subset X$ a convex, weakly compact and separable set. Let $x_{n}$ a sequence in $K$. Thus, given any $\epsilon >0$ there ...
2
votes
0answers
338 views

Fundamental theorem of calculus of Banach-space valued functions

Let $f:[a,b]\rightarrow E$ be a continuous function from the interval $[a,b]$ to a Banach space $E$. Let $F(x)=\int_a^xf(t)\text{ }dt$ where the integral is the Bochner integral. I have to prove that ...
5
votes
2answers
147 views

Prove that $\int_{D}\nabla u\cdot\nabla vdx=\int_{D}uv\,dx=0$

Let $D$ be the open bounded subset in $\mathbb{R}^{n}$ with smooth boundary, $\alpha$ and $\beta$ be different non-null real numbers, and $u$ and $v$ be in $W_0^{1,2}(D)\setminus\left\{ 0\right\} $ ...
2
votes
1answer
313 views

If the dual spaces are isometrically isomorphic are the spaces isomorphic?

Let $X$, $Y$ be Banach spaces such that the duals $X^\ast$ and $Y^\ast$ are isometrically isomorphic. Are $X$ and $Y$ necessarily isomorphic? The answer to the question whether $X$ and $Y$ are ...
0
votes
1answer
58 views

In which Banach spaces of functions do the convergence almost everywhere and boundedness of norms imply the weak convergence?

See the title. If I am not mistaken, this is true for $L^2$-spaces but is false for $L^1$-spaces. Say, should the space be reflexive?
2
votes
1answer
59 views

compact operators and finite dimentional spaces

Let $Q_n$ a finite dimentional space. Since any finite rank operator is compact, it's true that any linear operator $K:Q_n\to Q_n$ is compact 'cause $\dim(R(K))<\infty$?
3
votes
1answer
158 views

Extending continuous, densely-defined linear maps between locally convex spaces

Let $X$ and $Y$ be locally convex topological vector spaces, say over $\mathbb{C}$. To set the stage a bit, I'll say that the topology on $X$ is given by a separating family of semi-norms $(p_i)_{i ...
2
votes
1answer
53 views

Multiplicative operator from L1 to L1 is given by an L_inf function

Problem: Let $\phi :X\rightarrow \mathbb{C}$ be a measurable function with respect to a measure space $(X,\mu)$. Suppose that $\phi f\in L^1(X,\mu)$ whenever $f\in L^1(X,\mu)$ and define $M_\phi ...