# 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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### Why does one necessarily need the triangle inequality

I'm studying basic topological, metric and normed spaces and I am curious why one of the axioms of both a metric and a norm is the triangle inequality. It makes some sense to me having the triangle ...
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### Determining whether equality $\|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
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### distance between two eigen vectors corresponding to two different matrices in a normed space

Let $A$ and $B$ are two $n\times n$ matrices. Let 1) $Ax = \lambda x$ and 2) $By=\mu y$ for $x,y$ in a normed space. $\lambda, \mu$ are scalar. Also, for $x,y$ are unique eigen vectors (upto a ...
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### Variant of Holder's inequality: $\|x\|_p \le n^{\frac1p- \frac1r} \|x\|_r$

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
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### Tricky norm-inequality $\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$ for $p \in (0,1)$

For $r>p \ge 1$ one can show that in $\mathbb{C}^n$ we have $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ My question is now: Does this also hold for $1 \ge r>p>0$? Obviously we ...
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### Prob. 2, Sec. 2.8 in Kreyszig's functional analysis text: What is the norm of these bounded linear functionals on $C[a,b]$?

Let $C[a,b]$ denote the normed space of all the continuous (real or complex-valued) functions defined (and continuous!) on the closed interval $[a,b]$ on the real line, where $a, b \in \mathbb{R}$ and ...
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### Is $\mu A+\lambda B$ closed? [duplicate]

Let $E$ be a real normed space, $\mu>0, \lambda >0$ and $A,B \subset E$ closed and convex sets such that $0 \in A$ and $0\in B$. Is $\mu A+\lambda B$ closed? This is what I did so far: ...
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### Trouble finding a function satisfying an integral equation

I'm stuck at the last step of this exercise: b) Use the Banach fixed point theorem to show that there is a unique function $f \in C[0,1]$ for which the equation f(t) + \int_0^1e^{\tau+t-3}f(\tau)...
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### Isometric isomorphism between $R^2$ and $R$

Can someone help me solving the following problems? $(\mathbb R^2,d_2)$ and $(\mathbb R, d_1)$, $d_2, d_1$ being the respective euclidean norms, are not isometric isomorphic, i.e. there is no ...
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### Every Cauchy sequence in $\{f\in (C([0,1]),\|\cdot\|_1)\,|\,\exists a,b\in\mathbb R:f(x)=ax+b\}$ converges

I have trouble proving that, using the norm $\|f\|=\int_0^1|f(x)|\mathrm dx$, for a Cauchy sequence of functions $f_n(x)=a_nx+b_n$, the sequences $(a_n)_n$ and $(b_n)_n$ also have to be Cauchy ...
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### Problem regarding isometric isomorphisms [duplicate]

I need help regarding the following two exercises: a) Show that $(\mathbb R^2, d_2)$ and $(\mathbb R,d_1)$ $d_2,d_1$ being the respective euclidean metrics, are not isometric isomorphic, i.e. ...
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### Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...