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Tarea 1, Análisis real

Due August 14

Problem 1

The function $latex d_T(x,y) = |x^1 - y^1| + \ldots + |x^n - y^n|$ defined for $latex x,y\in\R^n$ is a metric on $latex \R^n$.

Problem 2

Two norms $latex ||\cdot||_1$ and $latex ||\cdot||_2$ on a vector space are equivalent if there exist constants $latex c_1, c_2$ such that

$latex c_1||x||_1 \le ||x||_2 \le c_2||x||_1$

for all $latex x\in X$.

  1. The norms $latex ||\cdot||_E, ||\cdot||_M$ and $latex ||\cdot||_T$ on $latex \R^n$ are equivalent.

  2. If $latex ||\cdot||_1$ and $latex ||\cdot||_2$ are equivalent and $latex B^i_r(x)$ is the ball of radius $latex r$ with center $latex x$ with respect to the metric induced by $latex ||\cdot||_i$, then, for each $latex \e>0$, there exist $latex \delta_1,\delta_2>0$ such that $latex B^1_{\delta_1}(x) \subset B_\e^2(x)$ and $latex B^2_{\delta_2}(x)\subset B_\e^1(x)$.

  3. Let $latex ||\cdot||_1$ and $latex ||\cdot||_2$ two norms on $latex X$, and suppose there exist $latex \delta, \e>0$ such that $latex B^1_\delta(0) \subset B_1^2(0)$ and $latex B_\e^2(0)\subset B_1^1(0)$. Then $latex ||\cdot||_1$ and $latex ||\cdot||_2$ are equivalent.

  4. Equivalent norms induce the same topology; i. e., $latex U$ is open in $latex (X,||\cdot||_1)$ if and only if it is open in $latex (X,||\cdot||_2)$.

Problem 3

Let $latex ||\cdot||_1$ and $latex ||\cdot||_2$ be norms on $latex \R^n$ induced by inner products. Then they are equivalent.

Problem 4

The uniform, $latex L^1$ and $latex L^2$ norms on $latex C([0,1])$ satisfy

$latex ||f||_1 \le ||f||_2 \le ||f||_u$

for all $latex f\in C([0,1])$. However, not two of them are equivalent.

Problem 5

Let $latex \bar B_\e(x)$ be the closed ball with center $latex x$ and radius $latex \e>0$, and $latex \overline{B_\e(x)}$ the closure of the open ball $latex B_\e(x)$. Then $latex \overline{B_\e(x)} \subset \bar B_\e(x)$. Give an example where $latex \bar B_\e(x) \not\subset \overline{B_\e(x)}$.