A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
84
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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 ...
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
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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
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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 ...
33
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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 ...
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 $L^r$ ...
23
votes
2answers
550 views
How slow/fast can $L^p$ norm grow?
This is actually an exercise in Rudin's Real and Complex Analysis, $L^p$ spaces chapter. Could anyone help me out? Thanks in advance.
Motivation: It's well known that if we have a function $f$ which ...
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 ...
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
769 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
573 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 + ...
16
votes
4answers
645 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 ...
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
0answers
655 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
4answers
297 views
Banach spaces over fields other than $\mathbb{C}$?
Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...
12
votes
1answer
398 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
614 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 ...
12
votes
2answers
521 views
isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?
Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$.
Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
12
votes
1answer
631 views
Operator norm and tensor norms
I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by
$$
...
12
votes
1answer
128 views
Supremum of Banach Spaces
Let $X$ be a linear space with a family of complete norms $(\| \circ \|_I)_{I \in \mathcal{I}}$ on $X$, i.e. for every $I \in \mathcal{I}$ the tuple $(X,\|\circ\|_I)$ is a Banach space. Now define ...
11
votes
1answer
410 views
Different versions of Riesz Theorems
In Wikipedia, there are three versions of Riesz theorems:
1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space;
2 The representation theorem for ...
11
votes
1answer
322 views
Renorming $\mathcal{B}(\mathcal{H})$?
Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
11
votes
1answer
387 views
Nested sequences of balls in a Banach space
This seems to be a fairly easy question
but I'm looking for new points of view on it
and was wondering if anyone might be able to help
(by the way- this question does come from home-work,
but i've ...
11
votes
1answer
431 views
Does separability follow from weak-* sequential separability of dual space?
Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a ...
11
votes
1answer
640 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 ...
11
votes
1answer
79 views
Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$
Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$.
I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
11
votes
1answer
199 views
Banach spaces isomorphic to square
This is another exercise from Allan's book "Introduction to Banach Spaces and Algebras".
Exercise 2.9: A Banach space $E$ is said to be homeomorphic to its square if $E\oplus E$ is linearly ...
11
votes
1answer
452 views
Are the coordinate functions of a Hamel basis for an infinite dimensional Banach space discontinuous?
The question is in the title really, but I suppose I could at least fix some notation here.
Let $X$ be an infinite-dimensional Banach space - over the reals for the sake of concreteness. Use choice ...
10
votes
1answer
258 views
A Hamel basis for $l^{\,p}$?
I am looking for an explicit example for a Hamel basis for $l^{\,p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one ...
10
votes
1answer
280 views
Where does the theory of Banach space-valued holomorphic functions differ from the classical treatment?
For a Banach space $V$ over $\mathbb{C}$ and $U \subset \mathbb{C}$ open, one can easily check that the notions of holomorphy hold for maps $f: U \rightarrow V$ just as in the classical sense. Indeed, ...
10
votes
2answers
463 views
The space of Riemannian metrics on a given manifold.
For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set
$$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$
I'd like to know about the typical ...
10
votes
1answer
317 views
Proof of Hölder inequality by differentiation
I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...
9
votes
2answers
174 views
If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?
Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space.
Using this norm it's easy to show that if ...
9
votes
1answer
846 views
Weak-to-weak continuous operator which is not norm-continuous
Can one give a "relatively easy" example of a linear mapping $T\colon X\to X$ ($X$ a Banach space) which is
a) weak-to-weak continuous
b) weak*-to-weak* continuous ($X=Y^*$)
but not norm-to-norm ...
9
votes
1answer
224 views
On the norm of a quotient of a Banach space.
Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm
$$
\left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
9
votes
2answers
469 views
Applications of the Hahn-Banach Theorems
Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)?
Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my ...
9
votes
1answer
150 views
If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?
Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive?
To prove the contrapositive, it will suffice to assume that $X$ ...
9
votes
1answer
155 views
Growth $\beta X\setminus X$ of a Banach space $X$
Is there an analytic characterisation of the Cech-Stone compactification (in the norm topology, which is a normal space) of a Banach space $X$? The reason I ask is because I want to know what the ...
8
votes
3answers
334 views
Is there an easy example of a vector space which can not be endowed with the structure of a Banach space
Let $V$ be a real vector space.
Is there always a norm on $V$ such that $V$ is complete with respect to this norm?
If not, is there an easy counterexample?
8
votes
1answer
628 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 ...
8
votes
1answer
213 views
Every multiplicative linear functional on $\ell^{\infty}$ is the limit along an ultrafilter.
It is well-known that for any ultrafilter $\mathscr{u}$ in $\mathbb{N}$, the map\begin{equation}a\mapsto \lim_{\mathscr{u}}a\end{equation} is a multiplicative linear functional, where ...
8
votes
1answer
390 views
Space of Complex Measures is Banach (proof?)
How can we prove that the space of Complex Measures is Complete? with the norm of Total Variation.
I have stuck on the last part of the proof where I have to prove that the limit function of a Cauchy ...
8
votes
1answer
119 views
Criterion for convergence of the sequence of powers of a linear operator to $0$
Let $T$ be a linear operator in a Banach space $\mathbf{B}$. Suppose that for every $x \in \mathbf{B}$ there exists some real numbers $c_x>0$ and $a_x<1$ such that $||T^nx|| \leq ca^n$, for all ...
8
votes
1answer
157 views
Linear contraction on a Banach space
Let $X$ be a Banach space with a norm $\|\cdot\|_1$ and $A$ be a linear operator on $X$ such that
$\|A\|_1\leq 1$;
$\|A^m\|_1<1$ for some $m\in \mathbb N$.
Is that true that there is an ...
8
votes
1answer
252 views
A convergence problem in Banach spaces related to ergodic theory
Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition.
$\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$
$\frac{1}{n}\lVert ...
8
votes
1answer
158 views
Monotone convergence to a fixpoint in a Banach space
Let $\mathscr X$ be a complete separable metric space and $\mathbb B$ be the Banach space of all real-valued bounded measurable functions on $\mathscr X$. The partial order on this space is introduced ...
8
votes
1answer
324 views
On Pitt's theorem
The famous Pitt's theorem asserts that if $p>q$ then each bounded operator $T\colon \ell^p\to\ell^q$ is compact. Since $\ell^p$ and $\ell^q$ are incomparable ($p\neq q$, $p,q\geq 1$), each operator ...
