※ 引述《dondonmike (我很單純)》之銘言：
: Let V be the real vector space of all function f from R to R
: prove or disprove following sets of functions are subspace of V
: 1. all f such that f(x^2)=f(x)^2
Let A={f in V: f(x^2)=[f(x)]^2 for all x}
設 f, g in A, a, b in R.
(af+bg)(x^2) = af(x^2)+bg(x^2)
= a[f(x)]^2 + b[g(x)]^2
≠{af(x)+bg(x)}^2 in general.
故 A 非 V 之 subspace.
: 2. all f such that f(0)=1
Let B={f in V, f(0)=1}
設 f, g in A, a, b in R.
則 (af+bg)(0) = af(0)+bg(0)
= a+b
in general ≠ 1
故 B 非 V 之 subspace.
以下自己做吧!
: 3. all f such that f(0)=1+f(1)
: 4. all f which are continues
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