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 $\left\|\sum_{n=0}^\infty x_n\right\|=\sum_{n=0}^\infty|x_n|$.

6435 questions
0
votes
2 answers

Please help me proving that the sum is harmonic

If ${c_n}$ is a bounded sequence, then $$ f(r, \theta)=\sum_{n=-\infty}^{\infty}c_nr^{|n|}e^{in\theta} $$ is harmonic in the disc. Help me proving?
Matema Tika
  • 869
0
votes
1 answer

Dense subspace in $l^{2}$

Let $(\lambda_{n})_{n=1}^{\infty}$ be a sequence of scalars nonzero, $S = \{x = (\xi_{j}) \in l^{2}: \sum^{\infty}_{j=1}|\lambda_{j}||\xi_{j}|<\infty\}$ and the opetator $T:S \rightarrow l^{2}$ defined by $Tx=(\lambda_{j}\xi_{j})$. Show that $S$ and…
T.S.A
  • 55
  • 5
0
votes
1 answer

nonempty closed convex subset of reflexive Banach space achieves its minimum norm

Let $X$ be a reflexive Banach space and $K$ a nonempty closed convex subset of $X$. prove that there exists an $x\in K$ such that $\|x\|=\inf\limits_{y\in K}\|y\|$. I try to prove it in the way that $X$ is a Hilbert space but I fail because…
user387147
  • 322
0
votes
1 answer

C(K) that contains $c_0$ complemented but $K$ does not contain convergent sequence.

Let $K$ be compact Hausdorff topological space, $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$, endowed with supremum norm. It is known that if $K$ contains convergent sequence then $c_0$ is complemented in $C(K)$. I…
Aligomez
  • 47
0
votes
1 answer

$C[X,R]$ a closed subspace of $B[X,R]$

Is there any condition for $X$ to be a compact space in the theorem "Space $C[X,R]$ is a closed subspace of $B[X,R]$? Because I see that if $X$ is not compact then $C[X,R]$ does not become a subset of $B[X,R]$. Counterexample is that $X=(0,1)$ is a…
Sultan Abdullah
  • 1
0
votes
0 answers

Contraction function.

I have a function $h\left(x\right)=\sqrt{\mu x + \eta}$ such that $\mu\in\left(0,+\infty\right)$ and $\eta\in\mathbb{R}\setminus\left\{0\right\}$. I want to check the domain and range of $h$ in case the coefficient $\eta<0$ or $\eta>0$. To do so,…
user572171
  • 41
0
votes
2 answers

Banach space convergence

Having $X$ , a Banach space. Show that $\{x_n\}$ converging to $x$ implies that for all functions $f$ contained in $X^\ast$ (dual), $f(x_n)$ converges to $f(x)$.
mimi
  • 47
0
votes
1 answer

Is the Banach space $M_{n\times n} (\mathbb{C})$ with normal structure?

Is the Banach space $M_{n\times n} (\mathbb{C})$ with normal structure? I know the Banach space $\oplus_{1}^{n} \mathbb{C}$ is with normal structure but I can't fine a subspace of $M_n(\mathbb{C})$ without nondiametral point.
Darman
  • 804
  • 5
  • 15
0
votes
1 answer

Every proper subspace of a Banach space is either closed or dense

Let $E$ be Banach, $F$ a proper subspace, take $y$ outside $F$, define $f(x+ty)=t$ for $x$ in $F$ and $t$ real, it is a linear functional and the kernel is $F$. Then $F$ not dense iff $F$ closed in $E$ iff $f$ is bounded. Is it correct to say I…
lucmobz
  • 363
0
votes
1 answer

Linear operator from a normed space to the set of all functions on $[0,1]$

Suppose that $X$ is a Banach Space and that $T: X \to C^n [0,1]$ is a linear operator. Assume that for any $t \in [0,1]$, any $k$ with $0 \leq k \leq n$ and any sequence $(x_j)^\infty\,_{j = 1}$ in $X$ with $\lim_j \|x_j| = 0$, $\lim_j (Tx_j)^k =…
taupi
  • 377
0
votes
1 answer

X^* closed under pointwise convergence.

The pointwise limit $\psi$ of a sequence $(\psi_n)$ in the topological dual $X^*$ of a Banachspace $X$ is of course linear. How can I show that it is continous, too?
durst
  • 162
0
votes
1 answer

Show that $(X,\|\cdot\|)$ is a banach space.

Let $g:[0,\infty)\to[1,\infty)$ a continuous strickly increasing function, with $g(0)=1$ and $g(t)\to\infty$, as $t\to\infty$. Let $X=\{x\in C([0,\infty)); \sup_{t\geq 0}{\dfrac{|x(t)|}{g(t)}}<\infty\}$. And if define, $$\|x\|=\sup_{t\geq…
user491599
0
votes
1 answer

Reflexivity of the intersection of Banach spaces

Let $X,Y$ be Banach spaces and denote $Z = X \cap Y$. It is easy to show that $Z$ is a Banach space with norm $$\|x\|_Z:= \|x\|_X+\|x\|_Y.$$ Assume that $X$ is reflexive and $Y$ is non-reflexive. Can we conclude anything about the reflexivity of…
Marry Mag
  • 539
0
votes
1 answer

Only Banach Spaces can be reflexive

Let $E$ be a normed space over $\mathbb K$. For each $x \in E$, let $x̂ : E^* \to \mathbb K$ be defined by $x̂ (l) = l(x)$, $l \in E^*$. Then the map $\Lambda_E: E \to (E^*)^*, \Lambda_E(x) = x̂ $ is an isometric linear operator. $E$ is called…
Pazu
  • 1,077
0
votes
1 answer

Show that the collection $\{ M_x:x\in S\}$ has finite intersection property

Currently I am reading the book 'Isometries in Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two', Chapter $7,$ page $6.$ Definitions: A $T$-set is a subset $S$ of a Banach space $X$ with the property that for any finite…
Idonknow
  • 15,643
1 2 3
…
8
9