Let V be the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of given in Exercises 12 through 15 are subspaces of V? The geometric sequences [i.e., sequences of the form , for some constants and K.
The geometric sequences [i.e., sequences of the form , for some constants and K are subspaces of V .
A subset W of a linear space V is called a subspace of V if
(a) W contains the neutral element 0 of V .
(b) W is closed under addition (if f and g are in W then so is f+g )
(c) W is closed under scalar multiplication (if f is in W and K is scalar, then kf is in W ).
we can summarize parts b and c by saying that W is closed under linear combinations.
Let us consider a set X to be the set of all geometric sequences which is a subset of V.
Consider two arbitrary elements from X namely x and y.
In a geometric sequence, on dividing two consecutive elements we get a common ratio. (i.e.)
Let us assume
Then, left side can be expressed as
Right side can be expressed as
It is obtained that L.H.S R.H.S. Thus, it is not closed under addition.
Therefore, it is not a subspace of V.
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