Tagged Questions

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

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Non linear compact map

Suppose to have two Banach spaces $E$ and $F$, with $E$ reflexive. Suppose to have a continuous map $T:E \to F$ which maps bounded subsets into precompact subsets. $T$ is not assumed to be linear. ...
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Existence of a basic sequence with basis constant $1$ in a Banach space

I am interested in a problem motivated by the following theorem. A proof (and the relevant definitions) can be found in many textbooks about Banach space theory, see for example Corollary 1.5.3 in ...
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In a normed space, the sum of a Closed Operator and a Bounded Operator is a Closed Operator.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following lemma Let $T:\mathcal{D}(T)\to Y$ be a bounded linear operator with domain $\mathcal{D}(T)\subset X$...
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239 views

Proving a particular function is surjective on a Banach space.

Let $(E,\|x\|)$ and let $f: E \to E$ such that $f+Id$ is a contraction ($Id$ is the identity map). Prove that $f$ is surjective and prove that, if $f$ is linear, then $f$ is a homeomorphism. The ...
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Every Banach space isomorphic to a subspace of C(X) (for some X)

The book of Douglas says on page 12: Theorem (Banach): Every Banach space $B$ is isometrically isomorphic to a closed subspace of $C(X)$ for some compact Hausdorff space X. Proof: Let X be $(B^*)_1$ ...
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202 views

Weaker Than The Weak Topology?

The weak topology on a Banach Space $E$ is defined to have sub-base consisting of open balls of the form $B_\alpha(x,r) = \lbrace y \in E : \vert \alpha(x-y) \vert < r\rbrace$ for each $x \in E$ ...
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251 views

Subspace isomorphic to a complemented subspace

I'll begin by writing down the definitions I'm using, to avoid confusion. Let $X$ be a Banach space and let $Y$ be a subspace of $X$. We say that $Y$ is complemented in $X$ if there exists a linear ...
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126 views

Weak* subseries convergent implies norm convergent?

Suppose $X$ is a Banach space and $X^{*}$ is separable. Suppose that $\sum x_{n}^{*}$ is a series in $X^{*}$ which has the property that every subseries $\sum x_{n_{k}}^{*}$ converges weak*. Show that ...