Page 9 - Vector Analysis

§1.3 Normed Vector Spaces 5

Proof. Properties 1 through 3 are obvious. We focus on proving property 5 first, and as we
will see, property 4 is a direct consequence of property 5.

Let α P F satisfy α(v, w) = |(v, w)|. Then |α| = 1. For all λ P R,

(λαv + w, λαv + w) = (λαv, λαv) + (λαv, w) + (w, λαv) + (w, w)
= λ2}v}2 + λα(v, w) + λα(v, w) + }w}2
= λ2}v}2 + 2λ|(v, w)| + }w}2 .

Since the left-hand side of the quantity above is always non-negative for all λ P R, we must
have

|(v, w)|2 ´ }v}2}w}2 ď 0

which implies property 5. To prove property 4, we note that

}v + w} ď }v} + }w} ô }v + w}2 ď (}v} + }w})2
ô (v + w, v + w) ď }v}2 + 2}v}}w} + }w}2
ô Re(v, w) ď }v}}w}

while the last inequality is valid because of property 5. ˝

Remark 1.17. The inequality in property 5 is called the Cauchy-Schwarz inequality.

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Definition 1.18. Let V, (¨, ¨) be an inner product space. A basis B of V is called orthog-
onal if u ¨ v = 0 if u, v P B and u ‰ v, and is called orthonormal if it is an orthogonal
basis such that }v} = 1 for all v P B.

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Definition 1.19 (Orthogoanl complement). Let V, (¨, ¨) be an inner product space over
scalar field F, and W Ď V be a vector subspace of V. The orthogonal complement of
W, denoted by WK, is the set

WK = ␣v P V ˇ (v, w) = 0 for all w P W( .
ˇ

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Proposition 1.20. Let V, (¨, ¨) be an inner product space over scalar field F, and W be a
vector subspace of V. Then WK is a vector subspace of V.