Question

For a fixed point c∈[a,b] the set ={∈[a,b]∣(c)=0} is a subset of [a,b] above [a,b] Show that properties (1) to (8) are satisfied with respect to addition and scalar multiplication of .

(1) For any , , ℎ∈(), +(+ℎ)=(+)+ℎ.
(2) For any , ∈(), +=+.
(3) If we define the function :→ℝ as (x)=0, then ∈() and for all ∈(), +=+=.
(4) For any ∈(), -∈() and also +(-)=(-)+=.
(5) For any ,∈() and ∈ℝ, (+)=+.
(6) For any ∈() and , ∈ℝ, (+)=+.
(7) For any ∈() and , ∈ℝ, ()=().
(8) For 1∈ℝ and any ∈(), 1･=.

Answer 1

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Final answer:

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