Material :Daly analysis Lemma 3.3.3. If X is a normed space and xn, x, Yn, y are vectors in X,then the following statements hold.(a) Uniqueness of limits: If xn→ x and xn - y, then x = y.(b) Reverse Triangle Inequality: |||x|| – |||| < ||x – y||(c) Convergent implies Cauchy: If xn → x, then {xn}nɛN is Cauchy.(d) Boundedness of Cauchy sequences: If {xn}nɛN is a Cauchy sequence, thensup ||xn|| < o.(e) Continuity of the norm: If xn(f) Continuity of vector addition: If x,→ x, then ||xn || → ||r||.→ x and yny, then xn+yn → x+y.(g) Continuity of scalar multiplication: If xn → x and cn → c (where cn, care scalars), then cnan → cx.Proof. We will prove one part, and assign the proof of the remaining state-ments as Problem 3.3.8.(d) Suppose that {xn}nɛN is Cauchy. Then, by applying the definition ofa Cauchy sequence with the specific choice e = 1, we see that there existssome N > 0 such that ||xm – xn|| < 1 for all m, n > N. Therefore, for alln > N we have||rn||||Ln – IN +IN|| < ||xn – ¤ || + ||æ|| < 1 + ||®N||-%3D3.3 The Induced Metric113Hence, for an arbitrary n we have||Tn|| < max{||x1||, ... |"N-1||, ||TN|| +1}.Thus {xn}neN is a bounded sequence.Since a normed space is a vector space, we can define lines, planes, andother related notions. In particular, if x and y are vectors in X, then the linesegment joining x to y is the set of all points of the form tx + (1 – t)y, where0

Name: Material :Daly analysis Lemma 3.3.3. If X is a normed space and xn, x,
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Description: Material :Daly analysis Lemma 3.3.3. If X is a normed space and xn, x, Yn, y are vectors in X,then the following statements hold.(a) Uniqueness of limits: If xn→ x and xn - y, then x = y.(b) Reverse Triangle Inequality: |||x|| – |||| < ||x – y||(c) C

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Lemma 3.3.3. If X is a normed space and xn, x, Yn, y are vectors in X, then the following statements hold. (a) Uniqueness of limits: If xn → x and xn → y, then x = y. (b) Reverse Triangle Inequality: ||||| – ||y||| < ||x – y|l. (c) Convergent implies Cauchy: If xn → x, then {xn}n€N is Cauchy.