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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
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Putting a Lower Bound On The Power Type Of Modulus Of Uniform Convexity

Take a Banach space $(X,\|\cdot\|_X)$ and define, on the interval $[0,2]$, the modulus of uniform convexity to be $\delta_X(\epsilon) = \inf\{1-\frac{\|x+y\|}{2}:\|x\|=\|y\|=1, \|x-y\|=\epsilon\}$. $X$ has modulus of convexity of power type $q$ if,…
Dizzy
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Does the approximation property pass to subspaces?

A Banach space $X$ has the approximation property (AP) if for any compact subset $K$ of $X$ and any $\epsilon>0$, there exists a bounded finite rank operator $R$ such that $\| x - R(x) \| < \epsilon$ for every $x \in K$. Question: If $M \subset X$…
Idonknow
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Sequence of dense sets in Banach spaces

Let $A_0 \supset A_1 \supset A_2 \supset \cdots$ - sequence of embedded Banach spaces and $B_0 \supset B_1 \supset B_2 \supset \cdots$ - suquence of linear spaces such that $B_i$ dense in $A_i$, it is true that $\bigcap_{k=0}^\infty B_k = 0…
Mykola Pochekai
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Lower Bound on Norm of Sum of Linearly Independent Vectors in a Banach Space

Suppose X is a Banach space and $x_1,...,x_n$ are linearly independent vectors. Can we find some sort of lower bound for $||\sum x_i||$? What about if we restict ourselves to normalised vectors?
Tim
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Is the range of this operator closed?

I think I am stuck with showing closedness of the range of a given operator. Given a sequence $(X_n)$ of closed subspaces of a Banach space $X$. Define $Y=(\oplus_n X_n)_{\ell_2}$ and set $T\colon Y\to X$ by $T(x_n)_{n=1}^\infty = \sum_{n=1}^\infty…
Ville
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Does an infinite dimensional Banach space always admit an infinite dimensional, separable subspace?

If that can't be achieved, what if the Banach space is reflexive?
user219967
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Dual of $C(K,X)$, for scattered $K$

Let $K$ be a compact, Hausdorff space and $X$ be a Banach space. By $C(K,X)$ we denote the Banach space of all continuous functions $f : K \to X$, equipped with the supremum norm: \begin{align} \|f\|_\infty = \sup_{k \in K}…
Vinícius Morelli
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Why does it mean for Banach space to be a locally convex topological vector spaces

A Banach space is simply a complete normed linear space. According to Wikipedia it is also a locally convex topological vector space. How does complete + normed + linear space translate into locally convex + topological vector space?
Shamisen Expert
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Asymptotically nonexpansive mapping that is not nonexpansive

Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive mappings if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ such that $\lim\limits_{n\to \infty}k_n=1$…
user62498
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Can you show this space isn't complete?

Let $X=C^0([0,1])$ and $||\cdot||:X\to\Bbb R$ be defined as $$||f||=\max_{x\in[0,1]}x^2|f(x)|.$$ Show that $||\cdot||$ isn't a Banach space. (I can't find any Cauchy sequence that does not converge. Can you find one?)
User
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The example of a mapping is homomorphic but not isomorphic

The example of two Banach space A,B such that the mapping T:A→B, is onto , one-to-one and homomorphic but not isomorphic i.e ∥T(x)∥≠∥x∥. I think there are two norm spaces, such that norm does not lead to the inner product.
ali f
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Density of a closed and convex subset of a Banach space.

There's this homework problem that I've been battling on-and-off for about a week now. I've managed to reduce it to the following lemma. Lemma. Let $X$ denote a Banach space, and consider closed and convex $C \subseteq X$ such that for all $x \in…
goblin GONE
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$1$-subsymmetric basis

Let $X$ be a Banach space with a $1$-subsymmetric basis $(e_i)$. I'm trying to understand why it is the case that for any $x = \sum_{i=1}^\infty x_i e_i \in X$, any strictly increasing sequence $(n_i) \subset \mathbb{N}$, and any $(\epsilon_i)…
ragrigg
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Is the domain of a closure operator close?

Let $X$ and $Y$ be two Banach spaces. Let $A:D(A)\subset X \to Y$ linear and continuous. Then $A$ is closable. Let $\overline{A}$ be the closure operator. My question is: is $D(\overline{A})$ close?
user96849
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Banach spaces and Normed linear spaces

Here's a theorem: A normed linear space X is a Banach space iff every absolutely convergent series in X is convergent. How is this possible? I need the proof.
user73535
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