3.6.5. Let X be a finite-dimensional vector space, and let B = {₁,..., ed} be a Hamel basis for X Then for each x EX there exist d unique scalars c₁(x),..., ca(x) such that x = k-1 Ck (x) ek. Define ||*||p Prove that: = ((\c₁(x)|P + ... + \ca(x)|P)¹/p, if1≤p<∞, [max{|c₁(x)\,...,|ca(x)|}, if p = ∞. (a) || ||p is a norm on X, (3.27)

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Material :Daly analysis
= {e1,.…..,ea}
Then for each x € X there exist
3.6.5. Let X be a finite-dimensional vector space, and let B
%3D
be a Hamel basis for X
unique scalars c1(x),..., ca(x) such that x =
Ek=1 Ck (x) ek. Define
(lcı (x)P + -..+ |ca(x)|P)*/", if 1<p<∞o,
max{lc1 (x)|,..., |ca(x)|},
(3.27)
if p= o.
Prove that:
(a) | - ||p is a norm on X,
Transcribed Image Text:= {e1,.…..,ea} Then for each x € X there exist 3.6.5. Let X be a finite-dimensional vector space, and let B %3D be a Hamel basis for X unique scalars c1(x),..., ca(x) such that x = Ek=1 Ck (x) ek. Define (lcı (x)P + -..+ |ca(x)|P)*/", if 1<p<∞o, max{lc1 (x)|,..., |ca(x)|}, (3.27) if p= o. Prove that: (a) | - ||p is a norm on X,
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