Mind-Warm-Ups Vector Spaces

Ultimate Space = S
si = arbitrary partition of S
Δsi = infinitesimal si, that is limit of si as it tends to zero.
pi = arbitrary point in S = Δsi

Problem 41: what is a normed linear space?
Ans: let X be a linear space over the field F where F is either the real field R or the complex field C.
A norm on X is a real-valued function whose value at x is denoted by ||x||, satisfying the following conditions for all x, y, in X and &lpha; in F:
(i)||x|| > 0 if x ≠ 0
(ii)||αx|| = |α|||x||
(iii)||x + y|| ≤ ||x|| + ||y||
A linear space X with a norm defined on it is called a normed linear space.

Problem 42: Consider a vector space with origin at O in which the vector u is represented by the arrow from O to P and the vector v is represented by the arrow from O to Q. The projection of
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Problem 43:
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Problem 44 :
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Problem 45 :
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Problem 46:

Problem 47:
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Problem 48:
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Problem 49:
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Problem 50:
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Peter Oye Sagay

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