# All Questions

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### Equivalence of two norms in complete space

Let $X$ be a vector space with two norms $\| \cdot \|_1$ and $\| \cdot \|_2$ such that $\| x \|_1$ $\leq$ $\| x \|_2$ for all $x \in X$. If $X$ is complete in both norms, prove they are equivalent. ...
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### Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
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### Show that a set is a ring

Let $R\not=\{0\}$ be a commutative ring with unity. Let $I$ be a prime ideal in $R$. Let $S=R-I=\{x\in R|x\not\in I\}$. Let $F$ be a field that contains $R$ as a subring with the same unity. Show that ...
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### String transformation

There are $n$ light bulbs place in circle and colored with Red, Green, Blue. After 1 second, from left to right, 2 consecutive bulbs which have different color will both change to extant color. ...
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### Topological equivalence of any norm on $\mathbb C^n$

In University I have been told that every norm on $\mathbb C^n$, for any $n\in\mathbb{N}$, is equivalent to every other such norm. I have a proof for this on any vector space on $\mathbb R$. Trouble ...
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### Probability [Sum of digits is even for a random number] [closed]

A number of 6 digit numbers is written down at random. Probability that the sum of the digits is an even number is? Please answer with explanation.