530 CHAPTER 19. HILBERT SPACES

Definition 19.3.3 Let {x1, · · · ,xn} ⊆ H, a Hilbert space. Define

G (x1, · · · ,xn)≡

 (x1,x1) · · · (x1,xn)...

...(xn,x1) · · · (xn,xn)

 (19.3.19)

Thus the i jth entry of this matrix is (xi,x j). This is sometimes called the Gram matrix. Alsodefine G(x1, · · · ,xn) as the determinant of this matrix, also called the Gram determinant.

G(x1, · · · ,xn)≡

∣∣∣∣∣∣∣(x1,x1) · · · (x1,xn)

......

(xn,x1) · · · (xn,xn)

∣∣∣∣∣∣∣ (19.3.20)

The theorem is the following.

Theorem 19.3.4 Let M = span(x1, · · · ,xn) ⊆ H, a Real Hilbert space where {x1, · · · ,xn}is a basis and let y ∈ H. Then letting d be the distance from y to M,

d2 =G(x1, · · · ,xn,y)G(x1, · · · ,xn)

. (19.3.21)

Proof: By Theorem 19.3.1 M is a closed subspace of H. Let ∑nk=1 αkxk be the element

of M which is closest to y. Then by Corollary 19.1.13,(y−

n

∑k=1

αkxk,xp

)= 0

for each p = 1,2, · · · ,n. This yields the system of equations,

(y,xp) =n

∑k=1

(xp,xk)αk, p = 1,2, · · · ,n (19.3.22)

Also by Corollary 19.1.13,

∥y∥2 =

d2︷ ︸︸ ︷∥∥∥∥∥y−n

∑k=1

αkxk

∥∥∥∥∥2

+

∥∥∥∥∥ n

∑k=1

αkxk

∥∥∥∥∥2

and so, using 19.3.22,

∥y∥2 = d2 +∑j

(∑k

αk (xk,x j)

)α j

= d2 +∑j(y,x j)α j (19.3.23)

≡ d2 +yTx α (19.3.24)