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, ...
84
votes
1answer
2k views
Does the open mapping theorem imply the Baire category theorem?
A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice.
On the other hand, the three ...
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 ...
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 ...
40
votes
3answers
895 views
Paul Erdos's Two-Line Functional Analysis Proof
Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes ...
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 ...
37
votes
2answers
537 views
Looking for a function such that…
There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is:
...
37
votes
1answer
867 views
Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?
(ZFC)
Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space.
Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $.
Define $\: \mathbf{B}_0 ...
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 ...
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 ...
31
votes
6answers
1k views
Why don't analysts do category theory?
I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.
Recently, ...
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 $L^r$ ...
29
votes
3answers
1k views
What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for?
Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic ...
24
votes
0answers
424 views
Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
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 ...
22
votes
3answers
1k views
Norms on C[0, 1] inducing the same topology as the sup norm
This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying.
Let $C[0, 1]$ denote the ...
22
votes
3answers
415 views
Is $\mathbb{R}^{\infty}$ homeomorphic to $\mathbb{R}^{\infty}\setminus\{0\}$?
Let $\mathbb{R}^{\infty}$ be a linear topological space of all sequences
$x=(x_{1},x_{2},\ldots,x_{n},\ldots)$ of real numbers with a product topology,
or, in other words, let $\mathbb{R}^{\infty}$ be ...
22
votes
2answers
432 views
Operators with finite spectrum
Suppose that $T$ is a bounded operator with finite spectrum. What happens with the spectrum of $T+F$, where $F$ has finite rank? Is it possible that $\sigma(T+F)$ has non-empty interior? Is it always ...
21
votes
4answers
706 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 ...
21
votes
2answers
791 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 ...
20
votes
3answers
907 views
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
20
votes
2answers
641 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?
20
votes
1answer
763 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 ...
20
votes
2answers
571 views
Is the fundamental theorem of calculus independent of ZF?
By the fundamental theorem of calculus I mean the following.
Theorem: Let $B$ be a Banach space and $f : [a, b] \to B$ be a continuously differentiable function (this means that we can write $f(x + ...
19
votes
3answers
504 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 ...
17
votes
2answers
506 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 ...
17
votes
1answer
377 views
Is $[0,1]^\omega$ homeomorphic to $D^\omega$?
Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case.
This observation leads to ...
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 ...
17
votes
3answers
576 views
When do weak and original topology coincide?
Let $X$ be a topological vector space with topology $T$.
When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset ...
17
votes
3answers
422 views
Prove an inequality on $l^2$ sequences over $F_2$
Denote $F_2$ the free non-abelian group on two letters $a, b$.
Note that any element in $F_2$ is just a word formed by letters from the set $\{a,b,a^{-1},b^{-1}\}$, and the group structure is given ...
16
votes
4answers
643 views
Example of different topologies with same convergent sequences
It's well known that for metric spaces the following is true
Let $ X $ be a space with two different metrics $ d_1,d_2$ such that the two topological spaces $ (X,d_1),(X,d_2) $ have the same ...
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 ...
16
votes
2answers
606 views
When is a notion of convergence induced by a topology?
I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...
16
votes
1answer
546 views
What is the spectrum of the commutative C*-algebra I have constructed here?
Let $B$ and $F$ be compact Hausdorff spaces.
Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$.
I think this induces a fiber ...
15
votes
3answers
217 views
To what extent is the taylor polynomial the best polynomial approximation?
Given a function $f\in\mathscr C^n([a,b])$ and a point $x_0\in [a,b]$, to what extent is the n-th taylor polynomial $T_n(x,x_0)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$ the best polynomial ...
14
votes
3answers
621 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, $S\subset{\bf C}$, ...
14
votes
3answers
715 views
Not every metric is induced from a norm
I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function
$d(u,v) = \lVert u - v \rVert$, $u,v \in V$.
My question is whether every metric ...
14
votes
2answers
333 views
Increasing orthogonal functions
What is the maximal $n$ such that there exist functions $f_1, \dots, f_n:[0,1] \to \mathbb{R}$ that are all bounded, non-decreasing, and mutually orthogonal in $L^2([0,1])$?
13
votes
1answer
1k views
Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?
the Fourier transformation of a scalar function with respect to one variable might be defined as
$\mathcal{F}\left[w\right](\omega )\equiv ...
13
votes
1answer
526 views
Medial Limit of Mokobodzki (case of Banach Limit)
A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
13
votes
3answers
1k views
Discontinuous linear functional
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, ...
13
votes
1answer
526 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) ...
13
votes
1answer
277 views
How does $\sigma(T)$ change with respect to $T$?
Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane.
I wonder whether there is some result concerning how ...
13
votes
2answers
379 views
Question about the Riesz representation theorem(s)
I am looking at two seemingly same (but not quite) Riesz representation theorems:
(Wikipedia) Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the space of compactly supported continuous ...
13
votes
3answers
323 views
Are commutative C*-algebras really dual to locally compact Hausdorff spaces?
Several online sources (e.g. Wikipedia, the nLab) assert that the Gelfand representation defines a contravariant equivalence from the category of (non-unital) commutative $C^{\ast}$-algebras to the ...
13
votes
0answers
650 views
Continuous projections in $\ell_1$ with norm $>1$
I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
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
1answer
397 views
Cardinality of a Hamel basis
What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
12
votes
1answer
510 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.
12
votes
3answers
611 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 ...
