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, ...
36
votes
3answers
4k views
Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ || \cdot || $.
It's not hard to show that if $|| \cdot || = \sqrt{\langle \cdot, \cdot \rangle}$ for some ...
4
votes
1answer
426 views
Limit of $L^p$ norm
Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
21
votes
2answers
787 views
The identity cannot be a commutator in a Banach algebra?
The Wikipedia article on Banach algebras claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is ...
12
votes
1answer
552 views
How to prove that an operator is compact?
Consider $T\colon\ell^2\to\ell^2$ an operator such that
$Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
11
votes
1answer
637 views
Strong and weak convergence in $\ell^1$
Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
8
votes
1answer
625 views
Nonnegative linear functionals over $l^\infty$
My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
9
votes
1answer
616 views
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ ...
7
votes
4answers
428 views
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrasingly weak -
I would like to show that, for a real number $p ...
2
votes
2answers
551 views
Prove that $X'$ is a Banach space
I'm taking a new course on functional analysis and meet with the following problem.
If $X$ is a normed space (not necessarily complete), then prove that $X'$ is a Banach space.
Definition: When the ...
23
votes
13answers
4k views
Good book for self study of functional analysis
I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic ...
41
votes
3answers
1k views
Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).
One can use interpolation to show that ...
16
votes
2answers
1k views
Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)
I am trying to understand the differences between
$$
\begin{array}{|l|l|l|}
\textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline
\text{metric}& \text{metric ...
14
votes
3answers
618 views
Can spectrum “specify” an operator?
Given a bounded operator $A$ on a Banach space $X$, one may find the spectrum $\sigma(A)\subset{\bf C}$.
Here are my questions:
Given some set in the complex plane, say, $S\subset{\bf C}$, ...
2
votes
1answer
552 views
Inequality between $\ell^p$-norms
Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$).
Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
19
votes
3answers
498 views
If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$
I'm trying to prove the following:
If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that ...
12
votes
1answer
509 views
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
Maybe I would have to use the Rademachers.
17
votes
4answers
1k views
Square root of a function (in the sense of composition)
There are some math quizzes like:
find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$
such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$
If such $\phi$ exists (it does in this example), $\phi$ can ...
4
votes
1answer
252 views
Operator whose spectrum is given compact set
Let $A\subset \mathbb{C}$ be a compact subset.
Since $A$ is compact and metric space, it is separable, say $\overline{\lbrace a_n\rbrace_{n=1}^\infty}=A$.
Let $\mathcal{l}^2(\mathbb{Z})$ be the ...
50
votes
16answers
2k views
Your favourite application of the Baire category theorem
I think I remember reading somewhere that the Baire category theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
20
votes
2answers
640 views
Norm for pointwise convergence
Does there exist a norm on the space of all real-valued functions on the real line (or on an open set? a compact set?) such that convergence in this norm is equivalent to pointwise convergence?
11
votes
5answers
981 views
Does there exist a linear independent and dense subset?
Do there exist in infinitely dimensional normed spaces linearly independent and dense subsets?
(Existence of linearly independent dense subset is equivalent of existence of dense Hamel Basis.)
...
12
votes
2answers
2k views
The Duals of $l^\infty$ and $L^{\infty}$
Can we identify the dual space of $l^\infty$ with another "natural space"?. If the answer yes what about $L^\infty$. By the dual space I mean the space of all continuous linear functionals.
12
votes
3answers
610 views
Compactness of a bounded operator $T\colon c_0 \to \ell^1$
Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact.
I know how to prove this in case $\ell^r \to ...
8
votes
2answers
769 views
Finding the adjoint of an operator
This is from my homework, I'm totally lost as to how to proceed.
Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by
$(Tf)(x) = \int^x_0 f(s) \ ds$
What is the adjoint of $T$?
...
4
votes
1answer
1k views
“Every linear mapping on a finite dimensional space is continuous”
From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector ...
21
votes
4answers
704 views
Is Banach-Alaoglu equivalent to AC?
The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem.
As I have ...
3
votes
1answer
450 views
Are isometric normed linear spaces isomorphic?
I should know the answer to this (and I did some time ago, but have forgotten): If the normed linear spaces $X$ and $Y$ are isometric (there is a bijective map from $X$ to $Y$ that preserves ...
7
votes
4answers
3k views
Space of bounded continuous functions is complete
I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete.
The lecturer then proved two things, (i) that $f(x) = \lim ...
13
votes
1answer
525 views
Is there a constructive proof of this characterization of $\ell^2$?
I would like to revisit this question, which can be equivalently stated as:
Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
4
votes
2answers
874 views
Spectrum of Indefinite Integral Operators
I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities.
For the first, suppose $T:L^{2}[0,1]\rightarrow ...
2
votes
1answer
438 views
Question about Fredholm operator
$X,Y$ are Banach spaces and $A\in B(X,Y)$ is a Fredholm operator (that is, the dimensions of ker($A$) and coker($A$) are both finite), then are closed linear subspaces ker($A$) and Im($A$) ...
33
votes
1answer
2k views
Was Grothendieck familiar with Stone's work on Boolean algebras?
In short, my question is:
Was Grothendieck familiar with Stone's work on Boolean algebras?
Background:
In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
6
votes
0answers
264 views
On the weak and strong convergence of an iterative sequence
I have some difficulties in the following problem.
I would like to thank for all kind help and construction.
Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
votes
5answers
439 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}\}.$$
According to the book [Dunford and ...
6
votes
1answer
312 views
The $L^1$ convergence of $f(x - a_n) \to f(x)$
I'm trying to solve something, and I'm stuck.
Suppose you have a function $$f \in L^1(\mathbf R) $$ and a sequence of real numbers that converges to zero: $$ a_n \rightarrow 0 $$
define a sequence ...
4
votes
2answers
269 views
How to prove that $C^k(\Omega)$ is not complete
Let $\Omega \subset\mathbb{R}^n$ be some bounded domain. And Consider the set of all k-times differentiable functions $C^k(\Omega)$.
I want to prove that this set is not complete with the inner ...
0
votes
1answer
217 views
Weak *-topology of $X^*$ is metrizable if and only if …
Let $X$ be a topological vector space on which $X^*$ separates points. Prove that "the weak *-topology of $X^*$ is metrizable if and only if $X$ has a finite or countable Hamel basis"?
(A set $\beta$ ...
7
votes
2answers
370 views
Complement of $c_{0}$ in $\ell^{\infty}$
How can I show that $c_{0}$ cannot be complemented in $\ell^{\infty}$?
Complement in the following sense
$$c_{0}+V = \ell^{\infty}$$
And the projections are continuous.
6
votes
1answer
307 views
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $.
Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
3
votes
1answer
345 views
Spectrum of the operator
Let $T$ be an operator on Hilbert space.
Define $\sigma(T)=\lbrace \lambda\in \mathbb{C} | \lambda I - T~\textrm{is not invertible}\rbrace$.
How can I prove that $\sigma(T^n)=\lbrace ...
3
votes
1answer
283 views
$\omega$ - space of all sequences with Fréchet metric
I'm working on to prove the following:
Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence.
Any hint is ...
2
votes
1answer
348 views
A vector without minimum norm in a Banach space
Question:
Let $E = C[0, 1]$, with sup norm. Let $K$ consist of all $f$ in $E$ such that
$$\int_{0}^{\frac{1}{2}}f(s)ds-\int_{\frac{1}{2}}^{1}f(s)ds=1$$
Prove that $K$ is a closed convex subset of ...
6
votes
2answers
167 views
Representation of a linear functional in vector space
In the book Functional Analysis,Sobolev Spaces and Partial Differential Equations
of Haim Brezis we have the following lemma:
Lemma. Let $X$ be a vector space and let
$\varphi, \varphi_1, \varphi_2, ...
4
votes
0answers
148 views
An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? [duplicate]
Possible Duplicate:
Nonnegative linear functionals over $l^\infty$
Setup: Let $l^\infty$ be the set of bounded sequences (with terms in $\mathbb{R}$), and let $l^1$ be the set of sequences ...
40
votes
1answer
1k views
Example of a compact set that isn't the spectrum of an operator
This question is a follow-up to this recent question and related to that one.
Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset ...
31
votes
3answers
637 views
Instructive proofs in functional analysis
I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
30
votes
2answers
1k views
Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?
Is there an explicit isomorphism between $L^\infty[0,1]$ and
$\ell^\infty$?
In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
17
votes
2answers
505 views
Compact sets as point spectrum of a bounded operator
It is well known that if $K$ is any compact set in $\mathbb{C}$, then there exist a bounded linear operator $T:l_2\to l_2$ such that $\sigma(T)=K$. My questions are:
Q1) Does there exist $T$, a ...
5
votes
2answers
813 views
How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$?
Statement: If $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$ and vice versa.
One way of the proof.
We have $B(B^{-1}A ) B^{-1} = AB^{-1}. $ Assuming $ ...
20
votes
1answer
757 views
Are these two Banach spaces isometrically isomorphic?
Let $c$ denote the space of convergent sequences in $\mathbb C$, $c_0\subset c$ be the space of all sequences that converge to $0$. Given the uniform metric, both of them can be made into Banach ...
