# Tagged Questions

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### Understanding the right assumption about Gateaux derivative in Banach spaces

It is a general fact that the notion of Gateaux derivative is not uniform over the mathematical community i.e. someone requires it to be linear and continue other not and require this additional ...
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### a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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### Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
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### Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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### Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
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### Density of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}(\mathbb R)$

I am looking for a counterexample to $C^{1}_{0}(\mathbb R)$ ( $C^1$ functions with compact support) is dense in $L^{\infty}(\mathbb R)$? Is there some easy counterexample showing that this latter is ...
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### Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
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### The dual space of normed vector space $X$ is isomorphic to the dual of its completion

Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic. Why these are isomorphic? I don't know how to prove ...
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### Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...
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### existence of a weakly cauchy sequence if the dual space is separable [closed]

Let X be a normed space such that $X^*$ is separable. Given any sequence $(x_n)\in X$ then there exist a subsequence weakly cauchy
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### Isometry from Banach Space to a Normed linear space maps

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$. Let me have some idea to solve this. Thank you for your help.
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### Proving that $X/M$ is a banach space when $X$ is

I am trying to do an exercise in an introduction to functional analysis course: 1) Let $X$ be a normed space and $\{x_{n}\}_{n=1}^{\infty}\subseteq X$. Prove that $X$ is a banach space iff ...
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### Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...
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### Equivalent conditions for weak and weak-$*$ convergence

Say $E$ is a normed space over the field $\mathbb K$ ($\mathbb R$ or $\mathbb C$) and $E^{*}$ its dual space. The notations for weak and weak - $*$ convergence are $x_{n} \xrightarrow{w} x$ and ...
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### Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
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### A metric space $(\Bbb R,d)$ with $d(x,y)=||x-y||$ is complete!

I would like to receive only the hint, how to prove the statement on the heading. I understand that we have to prove that all Cauchy sequences converges in the space $\Bbb R$, e.g. ...
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### A trouble about the Ekeland variational principle

I have a trouble in the proof to $EVP$ theorem: About the existence of the $\lim (\varphi(y_n))$ ? Any hints would be appreciated.
### If $K$ is $w$−compact and convex, $f\in X^\ast \implies f$ attains its maximum on $K$
Let $X$ be a real Banach space If $K\subset X$ is weakly compact and convex, then for a given $f\in X^\ast$ (dual space) we can always find $k\in K$ such that \displaystyle \sup_{x\in ...