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1. Are the following statements true or false? Justify your answer with the help of a short proof or a counter example. (a) Every subspace of a Banach space is Banach. (b) On a normed space X, the norm function |I-|| :X→R is linear. (c) If T, and T2 are positive operators on a Hilbert space H, then T1+T2 is a positive operator on H.

1. Are the following statements true or false? Justify your answer with the help of a short proof or a counter example. (a) Every subspace of a Banach space is Banach. (b) On a normed space X, the norm function |I-|| :X→R is linear. (c) If T, and T2 are positive operators on a Hilbert space H, then T1+T2 is a positive operator on H.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Are the following statements true or false? Justify your answer with the help of a short proof or a counter example. 1. (a) Every subspace of a Banach space is Banach. (b) On a normed space X, the norm function ||-|| :X→R is linear. (c) If T, and T2 are positive operators on a Hilbert space H, then T, +T2 is a positive operator on H. (d) A closed map on a normed space need not be an open map. (e) Every finite dimensional normal space is reflexive.
Transcribed Image Text:Are the following statements true or false? Justify your answer with the help of a short proof or a counter example. 1. (a) Every subspace of a Banach space is Banach. (b) On a normed space X, the norm function ||-|| :X→R is linear. (c) If T, and T2 are positive operators on a Hilbert space H, then T, +T2 is a positive operator on H. (d) A closed map on a normed space need not be an open map. (e) Every finite dimensional normal space is reflexive.
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