# Questions tagged [normed-spaces]

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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### Let $m\in C[a, b]$. Consider on $(C[a, b], ||m||_∞)$ the Multiplication operator $A:C[a, b]\to C[a, b], Af=mf$. prove that $||A||=||m||_∞$.

Let $m\in C[a,b]\;$. Consider on $\big(C[a,b],\Vert m\Vert_{\infty}\big)$ the multiplication operator $\;A:C[a,b]\to C[a,b]\;,\;Af=mf\;$. Prove that $\;\Vert A\Vert=\Vert m\Vert_{\infty}\;$. My try: ...
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### Triangle inequality in norm [closed]

Let $X=\Bbb R^3$. For $x=(x(1),(x(2),x(3))\in X$, let $$||x||=[(|x(1)|^2+|x(2)|^2)^{3/2}+|x(3)|^3]^{1/3}$$. Then $||~||$ is a norm on $\Bbb R^3$. \begin{align*}||x+y||&=[(|x(1)+y(1)|^2+|x(2)+y(2)|^...
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### If a function is separately differentiable is it diferentiable?

Let $X$, $Y$ and $Z$ normed vector spaces $$f:X \times Y \to Z$$ Such that $f(x, \cdot)$ is differentiable for all $x \in X$ and $f (\cdot ,y)$ is differentiable for all $y \in Y$. Is $f$ ...
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### Let $f:\mathbb{E} \rightarrow \mathbb{F}$,where $\mathbb{E},\mathbb{F}$ are normed space,such that:

(I) $f(x+y)=f(x)+f(y).$ (II) $f$ is bounded in $B(0,1)$. Then $f \in L(E,F)$ My attempt: I proved $f (q x) = q x$ for all $x \in E$ y $q \in \mathbb{Q}$. Now I want to prove that $f (r x) = r f (x)$ ...
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### How to define the uniform norm on functions of $\mathbb{R}^n$?

I am unsure on how to apply the uniform norm on the following problem. Exercise 2.10 Let $C_0\big(\mathbb{R}^n\big)$ be the closure of the space $C_c\big(\mathbb{R}^n\big)$ of continuous, compactly ...
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### Is $\operatorname{Lip}_{\beta}[0,1]$ is closed subset in $\operatorname{Lip}_{\alpha}[0,1]$ for $0 < \alpha < \beta \leq 1$ ??

$\DeclareMathOperator{\Lip}{Lip}$ I already proved that if $0 < \alpha < \beta \leq 1$, then $\Lip_{\beta}[0,1] \subset \Lip_{\alpha}[0,1]$. Recall that we say that $f \in \Lip_{\alpha}[0,1]$ ...
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