MTH401
Question: Which of the following inequalities is best applicable in the proof of the Minkowski’s inequality?
Answer: Holder’s inequality.
Question: (1.) For arbitrary \\( x=(x_1,x_2,…〖,x〗_n)\\) and \\( y=(y_1,y_2,…〖,y〗_n)\\) in R^n , the inequality /( ∑_(i=1)^n▒〖|x_n y_n |≤∑_(i=1)^n▒〖(|x_n |^2 )^(1/2).∑_(i=1)^n▒(|y_n |^2 )^(1/2) 〗〗 /) is called
Answer: Cauchy Schwartz inequality
Question: Let (E, d) be a metric space. the expression /( K(x_0 ,r)={y∈E∶d(x_0,y)=r} /) best describes
Answer: the sphere, centre /( x_0 /) and radius r > 0
Question: (6.) Given a nonempty set E and /( x,y,z ∈E /) . Let the real-valued function f satisfy the conditions of a metric on E. Then one of the following is called a metric space:
Answer: (E, f)
Question: Let E be a nonempty set and d be the function /( d(x,y)={█(k>0, if x≠y@0,if x=y)┤ /) . The metric d on E is called
Answer: discrete metric
Question: Let (E, d) be a metric space and /( x_0 ∈E /) then the set /(B(X_0 ,r)={y ∈E∶d(x_0 ,y)<r} /) is called
Answer: an open ball
Question: In any metric space (E, d), each open ball is
Answer: an open set in E
Question: Let (E, d) be a metric space, a finite intersection of open sets in E is
Answer: in E
Question: (2.) Let E be an arbitrary nonempty set. Then /( d_(0 ):E×E→R /) defined by /( d_0 (x,y)={█(1, if x≠y@0,if x=y)┤ /) is a metric, called
Answer: discrete metric.
Question: Given the elements x and y in the set of real numbers R and let /( |.| /) denote the absolute value of numbers such that /( (R,|.| ) /) is a metric space. Which of the following is true?:
Answer: d│x-y│≥ 0
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