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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Connection between Fréchet derivative and the directional derivative in finite euclidean space

In the lecture notes I am reading, the following statement is made: Let $U$ be an open subset of $R^n$, and define the function $e:U \to R$. $e$ is said to be differentiable if for every $u \in U$ ...
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13 views

Minimum value of combined functions

Let $F(x) = f_1(x) + f_2(x) + f_3(x)$ If the minimum value of $f_1$ takes place when $x= x_1$, the minimum value of $f_2$ takes place when $x= x_2$, the minimum value of $f_3$ takes place when $x= ...
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21 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...
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34 views

Closed subsets of Banach space

$X=\{(a,b)\mid a \in C[0,1],b \in C[0,1]\}$, and its norm is $\|(a,b)\|=\|a\|_\infty+\|b\|_\infty.$ $Y=\{(a,a')\mid a \in C^1[0,1], \ a'(t)=\frac{da}{dt} \},\ Z=\{(0,b)\mid b \in C[0,1]\}.$ ...
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28 views

Example of a well defined functional integral?

So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define $$ \int_{f \in S} L(f) $$ As the sum of of every function $f$ evaluated by ...
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31 views

Properties of 1-Sphere in a linear normed space against a normed linear subspace.

Hi i have a little problem in understanding the proof of the following theorem If $N$ is a finite-dimensional proper subspace of normed linear space $X$, there exists an element in the 1-sphere of ...
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15 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator between the ...
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1answer
27 views

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
26 views

C*-Algebra: Cyclic Elements

Given a locally compact Hausdorff space $\Omega$. Consider the C*-algebra: ...
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1answer
11 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
66 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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15 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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25 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
25 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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30 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 ...
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1answer
18 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
24 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
32 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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33 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
22 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 ...
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83 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
31 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
19 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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43 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} * ...
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3answers
57 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
34 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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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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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 ...
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1answer
36 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 ...
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23 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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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 ...
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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 ...
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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
62 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
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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
29 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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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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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 ...
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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 ...
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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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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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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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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 ...
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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
450 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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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 ...