Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

A Banach space, named after Stefan Banach (1892–1945) is a complete normed vector space: a (real or complex) vector space equipped with a norm such that every Cauchy sequence converges. For instance, $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the usual norm (or, for that matter, any norm) is a Banach space. Another example is the space $\ell^1$ of all absolutely convergent series of real or complex numbers, equipped with the norm $\left\|\sum_{n=0}^\infty x_n\right\|=\sum_{n=0}^\infty|x_n|$.

5724 questions
222
votes
1 answer

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 classical consequences of the Baire category theorem in…
t.b.
  • 74,358
  • 9
  • 254
  • 333
133
votes
4 answers

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 if a function is in two $L^p$ spaces, (e.g. $p_1$…
user1736
  • 7,933
  • 7
  • 50
  • 60
120
votes
1 answer

Continuous projections on $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $S=\mathbb{N}$. I've found one quite general…
Norbert
  • 54,366
  • 8
  • 104
  • 198
79
votes
2 answers

The Duals of $l^\infty$ and $L^{\infty}$

Can we identify the dual space of $l^\infty$ with another "natural space"? If the answer is yes, what can we say about $L^\infty$? By the dual space I mean the space of all continuous linear functionals.
omar
  • 2,275
  • 2
  • 17
  • 19
77
votes
4 answers

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 $f(x) = \lim f_n (x)$ is bounded and (ii) that $\lim f_n \in…
Rudy the Reindeer
  • 43,529
  • 24
  • 140
  • 302
76
votes
3 answers

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 space} & \text{complete space}\\ \text{norm} & \text{normed} &…
vonjd
  • 8,348
  • 10
  • 47
  • 74
70
votes
1 answer

Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
mintu
  • 867
  • 1
  • 7
  • 8
54
votes
1 answer

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 $K \subset \mathbb{C}$ that can't be the spectrum of a bounded operator…
t.b.
  • 74,358
  • 9
  • 254
  • 333
49
votes
1 answer

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 spaces. It can be shown that the dual spaces of them…
YZhou
  • 2,023
  • 14
  • 25
48
votes
2 answers

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each $q_n$ is an equivalence class induced by $Y$, I…
Metta World Peace
  • 6,105
  • 5
  • 30
  • 56
46
votes
1 answer

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 topology on the prime spectrum of a ring? I let myself get…
t.b.
  • 74,358
  • 9
  • 254
  • 333
45
votes
2 answers

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 $L^r$ and $\ell^s$ for $1 \leq r,s \lt \infty$, unless $r$…
t.b.
  • 74,358
  • 9
  • 254
  • 333
43
votes
1 answer

Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space.

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}\mid f \text{ differentiable with }f' \text{ continuous}\}$$ with the $C^1$-norm $$\lVert f\rVert := \sup_{a\leq x\leq b}|f(x)|+\sup_{a\leq…
RDizzl3
  • 3,139
  • 2
  • 22
  • 36
43
votes
1 answer

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 $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert y\right\Vert_E\mid y\in x+F\}. $$ Unfortunately…
Jyrki Lahtonen
  • 123,594
  • 24
  • 249
  • 599
42
votes
2 answers

When is the image of a linear operator closed?

Let $X$, $Y$ be Banach spaces. Let $T \colon X \to Y$ be a bounded linear operator. Under what circumstances is the image of $T$ closed in $Y$ (except finite-dimensional image). In particular, I wonder under which assumptions $T \colon X \to T(X)$…
shuhalo
  • 6,666
  • 3
  • 26
  • 64
1
2 3
…
99 100