# Tagged Questions

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### Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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### Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
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### Hahn-Banach via Hamel Basis

my question for tonight: Is there a proof for Hahn-Banach using a Hamel Basis? I know, the proof for existence of Hamel Bases uses already Axiom of Choice, but I'd like to apply this without refering ...
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### Existence of a transpose of a linear operator

Let $X,Y$ be two linear spaces and let $b:X\times Y\to\Bbb R$ be a bilinear map. For any linear operator $A:X\to X$ we can define its $b$-transpose acting on $Y$ by the system of the following ...
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### Finite dimensional quotient $\Rightarrow$ closedness?

Let $X$ be a Banach space and $V\subset X$ a subspace. Suppose that the quotient space $X/V$ is finite dimensional. Is $V$ then closed? In other words: if $V$ is not closed, then can $X/V$ be finite ...
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### An upper bound for $\|(\lambda-A)^{-1}\|$?

Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$. If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...
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### What is the precise definition of predual

How does one define "predual" and the surrounding notions? More specifically: Why must there be only one predual of $X$ when $X$ is a Banach space? What is the correct notion of similarity here ...
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### Immediate predecessor in a chain of subspaces

Let $\mathcal{C}$ be a chain of subspaces of a Banach space $\mathcal{X}$. For each $\mathcal{Y}\in\mathcal{C}$, define its immediate predecessor ...
Let $X$ be a Banach space and $\mathcal{C}\subset\mathcal{L}(X)$ be a collection of bounded linear operators. A subspace $Y$ is said to be almost invariant under $\mathcal{C}$ if for each ...