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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Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) ...
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1answer
23 views

C*-Algebra: Cyclic Elements

Given a locally compact Hausdorff space $\Omega$. Consider the C*-algebra: ...
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1answer
8 views

Equality in definition of dual space norm

In the definition of the dual space norm, the WP page makes the following statement: and I was wondering why going from the middle equality to the right equality was obvious?
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4answers
43 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
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1answer
12 views

Dense Operators: Spectrum

This thread is Q&A. Given a Banach space $E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: ...
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17 views

The convergence of different metrics on the same space

The following example is from my notes, and I would like clarification on some wider points connected to it, namely about extensions from what we understand metrics and metric spaces to be. It follows ...
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1answer
18 views

Spectrum of weighted shift operator

The Banach space considered is the following: $(l^{\infty}(\mathbb{Z}), \|\cdot\|_{*})$ with $\|x\|_{*}=\|(...,x_{-1},x_{0},x_{1},...)\|_{*}=|x_{0}|+\text{sup}_{k\neq 0}|x_{k}|$. Define $A$, an ...
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1answer
25 views

Changing of the limits of integration with the integral metric.

Consider the following sequence of functions, $$f_n(x) = \begin{cases} nx & \text{for $0\le x \le \frac1n$} \\ 1 & \text{for $x\ge \frac1n$} \end{cases}$$ And call to mind the integral ...
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27 views

Weak convergence, $L^{2}$

I have a question about weak convergence. Let $(S,\Sigma,m)$ be a measure space. $(f_{t})_{t>0}$ be a family of square integrable functions. (i.e. for every $t>0$, $f_{t} \in L^{2}(S;m)$) and ...
2
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1answer
15 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
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1answer
22 views

Reference for solving linear operator equations

I'm interested in solving an equation of the form $$ Ax = b $$ for some bounded linear operator $A: H_1 \mapsto H_2$ where $H_1, H_2$ are some Hilbert spaces. I've seen in this math.SE post in ...
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1answer
31 views

Continuous functional on the linear operator

Let $\Pi, \hat \Pi$ be two linear operators from $U$ to $V$. The norm-distance is defined as $$||\hat \Pi- \Pi||=\sup_{x\in U}\frac{||(\hat \Pi- \Pi)x||}{||x||}$$ Let us define a continuous bounded ...
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1answer
31 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
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1answer
21 views

The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
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1answer
18 views

Evaluating 'Constant' Term

suppose I have a pde $$u_{xt}(x,t)+u(x,t)u_{xx}(x,t)=h(t),\,\,\,\, x\in[0,\pi],\,\, t>0$$ for some unspecified function $h(t)$. This question is about finding what $h(t)$ is. Please, you may ...
3
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1answer
72 views

Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
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1answer
30 views

Direct Sum: Stone

Problem Given Hilbert spaces $\mathcal{H}_\alpha$. Consider Hamiltonians: $$H_\alpha:\mathcal{D}H_\alpha\subseteq\mathcal{H}_\alpha\to\mathcal{H}_\alpha:\quad H_\alpha=H_\alpha^*$$ And their ...
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1answer
18 views

Is the identity in unital, simple, purely infinite $C^*$-algebra always infinite?

I'd like to prove that the identity, $I$, of a unital, simple, purely infinite $C^*$-algebra is always an infinite projection. What I'm hoping is that the following is true: If $p$ in $\mathfrak{A}$ ...
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0answers
41 views

A doubt regarding derivative of convolution!!

In the following calculation: $\int_{\mathbb R^{d}} u_{o \epsilon} div (\phi) dx = \int_{\mathbb R^{d}} (u_{o} * \psi_{\epsilon}) div(\phi) dx = \sum_{i=1}^{d} \int_{\mathbb R^{d}} ( u_{o} * ...
3
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3answers
55 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
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1answer
31 views

Square root of a compact normal operator

Halmos expresses below problem in his book; Problem: If $A$ is a normal operator and if $A^n$ is compact for some positive integer $n$, then $A$ is compact. I have an example in my mind which I ...
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2answers
87 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
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1answer
19 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
3
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1answer
34 views

The same topologies

Let $L^1 (\mathbb{Z})$ be the space of all functions $f:\mathbb{Z}\rightarrow \mathbb{C}$ such that $\left\{\|f\|=\sum_{k\in \mathbb{Z}}|f(k)|<\infty\right\}$. Clearly, $L^1 (\mathbb{Z})$ is a ...
2
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0answers
21 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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0answers
15 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
3
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1answer
46 views

A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
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1answer
52 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
2
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1answer
36 views

Boundedness of an operator with kernel

Let $K(x,y)$ be measurable in $\mathbb{R}^2.$ Suppose there is a positive, measurable (w.r.t Lebesgue measure on $\mathbb{R}$) $w(x)$ and $A\geq 0$ such that $$\int_{-\infty}^\infty \vert K(x,y) ...
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1answer
61 views

Trace: Independence

Problem Given a Hilbert space $\mathcal{H}$. Consider an operator: $$A\in\mathcal{B}(\mathcal{H}):\quad\operatorname{Tr}|A|<\infty$$ Regard ONB's: ...
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2answers
52 views

Deny Lions Lemma

I am working through the finite element book by Ciarlet and am currently looking at the Deny Lion's Lemma (Theorem 3.1.1 p. 115). The Lemma essentially wants to show that $\inf_{p \in P_{k}}\Vert v ...
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2answers
27 views

Prove that orthonormalsystem is an orthonormalbasis

We have an orthonormalsystem in $L^2(0, 2\pi)$: $\{e^{ikx} : k \in \mathbb{Z}\}$. Now I want to show that it's also an orthonormalbasis. I thought the easiest way to do that would be to show that ...
3
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1answer
28 views

Weak compactness of a set of translates in $C_0(\mathbb{R})$

Let $f \in C_0(\mathbb{R})$. Consider the set of translates of $f$ $$ A = \{ f_t : t \in \mathbb{R} \}$$ where $f_t(x)=f(x+t), x\in \mathbb{R}$. I want to show that $A$ is relatively compact in the ...
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1answer
20 views

Is adjoint space different from dual space? [on hold]

For a linear space X , whether the dual space and adjoint space are the same or different?
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32 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
2
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1answer
38 views

Fundamental solution for a parabolic PDE with costant coefficents

as it is well known, the fundamental solution of the heat equation is the function $G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}}$, for all $t>0,x\in\mathbb{R}^n$. I wonder if exists (and ...
3
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0answers
39 views

Physical meaning of Rudin's equation in Hilbert space

Rudin's Functional Analysis, p. 334, Corollary of Theorem 13.10 says Corollary If $a\in H$ and $b\in H$, the system of equations $$-Tx+y=a$$ $$x+T^*y=b$$ has a unique solution with $x\in ...
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0answers
28 views

Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
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0answers
26 views

Continuity of translation property [duplicate]

Let $u \in L^{p}(U)$ where $1 \leq p \lt \infty$ & $U \subseteq \mathbb R^{n}$ . Define : $F : \mathbb R^{n} \to L^{p}(U) $ by $ F(y) := u(x+y)$ . Prove that: as a function of $y$ ; $F(y) $ is ...
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0answers
28 views

Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$

Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$, closure of $(c_{00},\|\cdot\|_2)$ is $\ell^2$ and closure of ...
3
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1answer
45 views

Question on $L^p$ spaces defining metric

First question here so really excited and hope you can help me, thanks! In my intro to functional analysis class we just now covered $L^p$ spaces and I was presented with this homework question: ...
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4answers
445 views

Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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0answers
26 views

Statement about the discrete (metric) space, and both an open and closed ball.

I have the following statement from my notes: "Let $(X,d)$ be the discrete space i.e. any non-empty set with the discrete metric ($d_d(x,y)=1$ for all $x\neq y$). Then, amazingly, $B_1(x)=\{x\}$, a ...
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1answer
52 views

$Hom(V,W)$ remains unchanged when norms of $V$ and $W$ are replaced with equivalent norms.

I was thinking about the following question from section 3.4 of Loomis and Sternberg's Advanced Calculus The fact that $Hom(V,W)$ is unchanged when norms are replaced by equivalent norms can be ...
3
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2answers
45 views

Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
3
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1answer
31 views

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying the given condition.

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying: 1) $f_n\to0$ pointwise; 2) $\gamma_nf_n\not\to0$, for all $\gamma_n\in\mathbb C$ such that ...
2
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1answer
29 views

One norm is stronger than but not equivalent to another norm

Let $X=C^{1}([0,1])$. For $x \in X$, let $$||x||_{0}^{'}=|x(0)|+||x'||_{\infty}$$ I need to show that $||.||_{0}^{'}$ is stronger than but not equivalent $||.||_{\infty}$. It is easy to see that ...
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1answer
28 views

Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
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0answers
17 views

Showing the space of $\beta$-operators between Banach lattices is a Banach space [on hold]

Let $E,F$ be a Banach lattices. A linear map $T:E \rightarrow F$ is called $\beta$-operator if $\lVert T\rVert_\beta < \infty$, where $$\lVert T\rVert_\beta := \sup \bigl\{ \bigl\Vert ...
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0answers
30 views

Is this function uniformly convex?

I am using the following definition of uniformly convex: A continuous functional $G:Y \to R$ is uniformly convex on a ball $$B(0,\delta):=\{y\in Y : \|y\| < \delta\}$$ if there exists a ...