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6.7 The Function Space C[a,b] 181

SECTION 6.6 PROBLEMS

In each of Problems 1 through 5, write u as a sum of a 6. Let S be a subspace of R . Determine (S ) .
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⊥ ⊥
vector in S and a vector in S .
⊥
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7. Suppose S is a subspace of R . Determine a relation-
⊥
1. S has orthogonal basis < 1,−1,0,0 >, < 1,1,0,0 > ship between the dimensions of S and S .
4
in R , u =< −2,6,1,7 >. 4
8. Let S be the subspace of R spanned by < 1,0,1,0 >
2. S has orthogonal basis < 1,0,0,2,0 >, < −2,0, and < 0,0,2,1 >. Find the vector in S closest to
5
0,1,0 > in R , u =< 0. − 4,−4,1,3 >. < 1,−1,3,−3 >.
3. S has orthogonal basis < 1,−1,0,1,−1 >, < 1,0, 9. Let S be the subspace of R spanned by < 1,1,
5
5
0,−1,0 >, < 0,−1,0,0,1 > in R , u =< 4,−1, −1,0,0 >, < 0,2,1,0,0 > and < 0,1,−2,0,0 >.
3,2,−7 >. Find the vector in S closest to < 3,0,0,1,4 >.
4. S has orthogonal basis < 1,−1,0,0 >, < 1,1,6,1 > 10. Let S be the subspace of R spanned by < 0,1,1,
6
4
in R , u =< 3,9,4,−5 >. 0,0,1 >, < 0,0,3,0,0,−3 >,and < 0,0,0,0,
5. S has orthogonal basis < 1,0,1,0,1,0,0 >, < 0,1, 0,4 >. Find the vector in S closest to < 0,1,1,
7
0,1,0,0,0 > in R , u =< 8,1,1,0,0,−3,4 >. −2,−2,6 >.
6.7 The Function Space C[a, b]
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We will extend the notion of a vector space from R to a space of functions. This will enable us
to view Theorem 6.8 as an approximation tool for functions as well as an introduction to Fourier
series and eigenfunction expansions in Chapters 13 and 15.
Let C[a,b] denote the set of all (real-valued) functions that are continuous on a closed
interval [a,b].If f and g are continuous on [a,b], so is their sum f + g, defined by
( f + g)(x) = f (x) + g(x).

Furthermore, if c is any real number, then cf , defined by
(cf )(x) = cf (x)
is also continuous on [a,b].
The zero function θ is defined by θ(x) = 0for a ≤ x ≤ b, and this is in C[a,b].
These operations of addition of functions and multiplication of functions by scalars have the
same properties in C[a,b] as addition of vectors and multiplication of vectors by scalars in R .
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In this sense C[a,b] has an algebraic structure like that of R , and we also refer to C[a,b] as a
vector space. In this space we continue to denote functions by upper and lower case letters, rather
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than the boldface we used for matrices and vectors in R .

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Many of the concepts developed for vectors in R extend readily to this function space. We
say that f 1 , f 2 ,··· , f n in C[a,b] are linearly dependent if there are numbers c 1 ,··· ,c n ,
not all zero, such that
c 1 f 1 + c 2 f 2 + ··· + c n f n = θ.

This means that

c 1 f 1 (x) + c 2 f 2 (x) + ··· + c n f n (x) = 0

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October 14, 2010 14:21 THM/NEIL Page-181 27410_06_ch06_p145-186