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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2answers
153 views
On $L^p$ and $\ell^p$
If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
7
votes
1answer
232 views
Pointwise convergence of continuous functions implies uniform convergence on some open subset
For some Banachspace $A$ we have a sequence of continuous functions $g_n:A\rightarrow \mathbb{R}$ pointwise converging to some $g:A\rightarrow\mathbb{R}$. Prove that for any $\epsilon>0$ there ...
7
votes
2answers
665 views
$T$ is continuous if and only if $\ker T$ is closed
Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show
that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker ...
7
votes
1answer
147 views
Boundary of invertibles in a normed algebra
A student and I are reading the book Introduction to Banach Spaces and Algebras, by Allan, and we're stuck. Exercise 4.5 says:
Let $A$ be a normed algebra with unit sphere $S$. Let $a\in A$. ...
7
votes
1answer
363 views
How to find an integral kernel for poisson's equation in the upper half plane
In our lecture we have shown that $\forall f \in L^2(\mathbb{R}^n_+) $ there is a unique $ u $ in the Sobolev space $ H^2(\mathbb{R}^n_+) $ satisfying $ -\Delta u = f. $
Now in our exercise sheet we ...
7
votes
2answers
237 views
Continuity of semigroups on $L^2$ and $L^1$: Is this simple proof correct?
Let $(X, \mu)$ be a $\sigma$-finite measure space, and $P_t$ a symmetric, Markovian, strongly continuous contraction semigroup on $L^2(X,\mu)$. (Markovian means that if $f \in L^2$ with $0 \le f \le ...
7
votes
2answers
60 views
Is there a smooth compact set between any two compact sets?
today I saw the following statement and of course believe that this is true but I don't know how to prove it rigorously (and neither do my colleagues).
Let $\Omega \subset \mathbb{R}^n$ be open and ...
7
votes
1answer
153 views
Weak convergence in $L^p$ and uniform convergence
I don't understand the last line of a proof (which is supposed to be obvious...), could you help me?
The context is the following. We have a bounded open set $U$ of $\mathbb{R}^m$, and a ...
7
votes
1answer
149 views
Practical applications of the $L^p$ norm when $p \neq 1,2,\infty$
I'm roughly familiar with the concept of $L^p$ norms -- what they represent and how they are computed -- though I am far from educated in functional analysis in general. For reasons that are more or ...
7
votes
2answers
321 views
Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$
I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$.
Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$.
1st derivative: ...
7
votes
2answers
628 views
Dual of Sobolev space $W^{1,p}(U)$ for $U$ an arbitrary subset of $\mathbb R^n$
this question may be shameful, but nevertheless I can't help myself.
Let $U \subset \mathbb R^n$ be arbitrary, in particular not the whole of the space itself. I wonder about the dual of the space ...
7
votes
1answer
61 views
*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$
I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form
$A\in ...
7
votes
2answers
675 views
convolution of a function with itself equals itself
In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed.
for (1), $f*f=f$ ...
7
votes
1answer
260 views
Heat equation in bounded domain
Let $\Omega \subseteq \mathbb{R}^n$, $n\geq 2$, be a bounded domain with boundary $\partial \Omega \subseteq C^2,v$ outer unit normal vector on $\partial \Omega$, $h \in L^2(\Omega)$.
Let $u \in ...
7
votes
2answers
376 views
Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
I have to show that the real numbers equipped with the metric
$ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space.
Certainly, I have to search for a cauchy sequence of real numbers ...
7
votes
1answer
100 views
When is a $*$-homomorphism between multiplier algebras strictly continuous?
The strict topology on the multiplier algebra $M(A)$ of a C*-algebra $A$ is that generated by the seminorms
$$ x\mapsto \| ax \|\qquad x\mapsto\| xa \| \qquad (x\in M(A), a\in A) $$
Whereas a ...
7
votes
1answer
124 views
Dual space of space of all smooth function
On the space $C^\infty(S^1,\mathbb R)$, for each $n\in \mathbb N$, define
$$p_N(\gamma)= \max\{|f^{(k)}(t): t\in S^1, k\leq N\}$$
Topology of all norms above define a metrizable locally convex ...
7
votes
1answer
139 views
Weak generator of Feller semigroup
Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in ...
7
votes
1answer
407 views
Sex, Crime and Functional Analysis?
Ever since a friend told me about this book titled Sex, Crime and Functional Analysis: Part I. Functional Analysis, I have been looking for a copy of this book, but in vain.
SO does this book ...
7
votes
1answer
234 views
Do the two limits coincide?
Let $a$ be a non negative (positive almost everywhere) weight in $L_{loc}^1(\Omega)$, $\Omega\subseteq\mathbb{R}^n$ is open. For $\varphi\in C_c^{\infty}(\Omega)$ define
$$
...
7
votes
1answer
226 views
A map on the unit ball
Let $B$ be the unit ball of $\ell^2(\mathbb{N})$, i.e. $B=\lbrace x\in \ell^2(\mathbb{N}): \|x\|\le 1\rbrace.$ For each $x=(x_1,x_2,\cdots)\in B$, let
$$f(x)=(1-\|x\|,x_1,x_2,\cdots).$$ Define ...
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221 views
+150
Hille Yosida theorem application
Disclaimer: pretty long and specific (contraction semi groups involved).
I have fourth order parabolic equation
$$
u_t + \Delta^2 u = 0
$$
on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
7
votes
0answers
80 views
Existence of a map in a Hilbert space
Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$.
Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
7
votes
0answers
75 views
Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis
I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
7
votes
0answers
155 views
A type of local minimum (2)
Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. ...
6
votes
5answers
440 views
How to show that this set is compact in $\ell^2$
Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$
According to the book [Dunford and ...
6
votes
2answers
1k views
Differentiating an Inner Product
If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
6
votes
1answer
350 views
Isometric to Dual implies Hilbertable?
Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
6
votes
2answers
479 views
Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?
Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ?
(Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
6
votes
1answer
585 views
Vector, Hilbert, Banach, Sobolev spaces
Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
6
votes
3answers
263 views
Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$
In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is ...
6
votes
1answer
440 views
How to deduce open mapping theorem from closed graph theorem?
These two theorems are equivalent but I can not figure out how to deduce the open mapping from the closed graph. Can anyone give a hint or some reference?
6
votes
1answer
333 views
Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?
When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
6
votes
1answer
311 views
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $.
Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
6
votes
2answers
981 views
Distinguishing between symmetric, Hermitian and self-adjoint operators
I am permanently confused about the distinction between Hermitian and self-adjoint operators in an infinite-dimensional space. The preceding statement may even be ill-defined. My confusion is due to ...
6
votes
2answers
261 views
Difference between $u_t + \Delta u = f$ and $u_t - \Delta u = f$?
What is the difference between these 2 equations? Instead of $\Delta$ change it to some general elliptic operator.
Do they have the same results? Which one is used for which?
6
votes
2answers
193 views
How can one show that the topology of convergence in measure is separable?
Let $X$ be a polish space equiped with the borel sigma-algebra and a probability measure $\mu$. How can one show that the set of all borel measurable functions $f:X\rightarrow R $ ($R$ being the real ...
6
votes
1answer
434 views
Do there exist closed subspaces $X$, $Y$ of Banach space, such that $X+Y$ is not closed?
I am looking for an example of two closed subspaces of a Banach space, such that their sum is not closed.
6
votes
3answers
261 views
$L^1$ space with values in a Banach Space
I have been reading a bit about the Bochner integral and now I'm wondering the following:
For the theory to be "nice", one would expect that
$$L^1([0, \tau], L^1([0, \tau])) \cong L^1([0,\tau] ...
6
votes
2answers
148 views
Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$
I'm learning for an exam and I'm surprised by the following statement that is given without proof or example:
Let $\Omega\subset\mathbb{R}^n$ be open, bounded and connected and let $f\in ...
6
votes
1answer
141 views
How to prove this inequality in Banach space?
In a normed space $(E,\lVert \cdot\rVert)$ space we have the following inequality:
$$\forall\, x,y\in E,\quad\|x\|^{2}-\|y\|^{2}\leq \lVert x-y\rVert\cdot \|x+y\|.$$
How can we prove it?
6
votes
1answer
313 views
The $L^1$ convergence of $f(x - a_n) \to f(x)$
I'm trying to solve something, and I'm stuck.
Suppose you have a function $$f \in L^1(\mathbf R) $$ and a sequence of real numbers that converges to zero: $$ a_n \rightarrow 0 $$
define a sequence ...
6
votes
3answers
402 views
How is the uniform boundedness principle compatible with this seemingly weak convergent sequence?
In showing that $(x_i\rightharpoonup x)\not\Rightarrow(x_i\to x)$ or similar noncorallaries, one frequently uses the counterexample
$$
(u_i)_{i\in\mathbb{N}}\in \ell^2\colon \quad u_i = ...
6
votes
2answers
567 views
Some basics of Sobolev spaces
Let $W^{m,p}(\Omega) = \{ f \in L^p(\Omega): D^\alpha f \in L^p(\Omega) \text{ for multi-indices } |\alpha| \leq m\}$, where $D$ denotes the weak derivative. Let $W_0^{m,p}$ denote the closure of ...
6
votes
1answer
878 views
Metric and Topological structures induced by a norm
While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm.
1) Is it possible to find two equivalent ...
6
votes
2answers
70 views
Two terms that I want to understand: weakest topology and jointly continuous (in the following context).
I was reading an article online, please help me to understand the following lines (in bold letters). -
Topological structure:
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
6
votes
2answers
658 views
Weak Convergence implies boundedness and componentwise convergence
Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
6
votes
2answers
130 views
A sequence of singular measures converging weakly* to a continuous measure
Can anyone provide a sequence of singular (w.r.t. Lebesgue measure) measures $\in\mathcal{M}([0,1])=C[0,1]^*$ converging $weakly^*$ to an absolutely continuous (w.r.t. Lebesgue measure) measure?
6
votes
1answer
536 views
Example of a closed subspace of a Banach space which is not complemented?
In this post, all vector spaces are assumed to be real or complex.
Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there ...
6
votes
2answers
214 views
Question about positive operators on a Hilbert space
I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
