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Kunze Linear Algebra
· Problem 8.2.15
Inner Product Spaces
Mathematics — Methods · inner-products-direct-sum
Problem
Suppose
V
=
W
1
⊕
W
2
V = W_1 \oplus W_2
V
=
W
1
⊕
W
2
and that
f
1
f_1
f
1
and
f
2
f_2
f
2
are inner products on
W
1
W_1
W
1
and
W
2
W_2
W
2
, respectively. Show that there is a unique inner product
f
f
f
on
V
V
V
such that (a)
W
2
=
W
1
⊥
W_2 = W_1^\perp
W
2
=
W
1
⊥
; (b)
f
(
α
,
β
)
=
f
k
(
α
,
β
)
f(\alpha, \beta) = f_k(\alpha, \beta)
f
(
α
,
β
)
=
f
k
(
α
,
β
)
, when
α
\alpha
α
,
β
\beta
β
are in
W
k
W_k
W
k
,
k
=
1
,
2
k = 1, 2
k
=
1
,
2
.
Related theory
Inner Products And Inner Product Spaces
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