# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Looking for a collection of applications of the Open mapping theorem

I am looking for various applications of the open mapping theorem (https://en.wikipedia.org/wiki/Open_mapping_theorem_(functional_analysis) ) other than only the closed graph theorem and that if two ...
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### Show that $E$ is closed.

Let $X$ be a normed linear space .Let $T_n$ be a sequence of continuous linear operators on $X$ such that $\sup_n \|T_n\|<\infty$. Let $E=\{x:T_n x$ is Cauchy$\}$. Show that $E$ is closed. My ...
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### Proof of a p-norm and Cauchy/Cauchy-Schwarz inequality generalization

I've started to study metric spaces recently and got intrigued with a demonstration which led me to a bunch of ideas and questions. First of all, I've proved the Cauchy inequality and then Cauchy-...
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### Convergent series which is not absolutely convergent in $(X,\|\cdot\|)$

Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges. (a) Give an example of ...
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### Functional Analysis(Normed Spaces)

Let $X$ be the space of all complex valued square Riemann- integrable functions on $[0,1]$ with $2$- norm. Define the map $F:X\to X$ by $F(u)=v$ with $v(t)=\int\limits_{0}^{t}{ u^2(s) ds}$, then ...
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### How to show that $BV[0,1]$ ( the space of all functions on $[0,1]$ of bounded variation ) is not complete under supremum norm?

How to show that $BV[0,1]$ ( the set of all functions of bounded variation ) is not complete under supremum norm , by explicitly constructing a Cauchy sequence which does not converge or by ...