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Euclidean Spaces, Scalar Product, Real, Bilinearity, Reflexivity, Positivity, Trace, Cauchy, Bunyakovsky, Schwartz, Inequality, Triangle, Linear Algebra, Lecture Notes, Andrei Antonenko, Department of Applied Math and Statistics, Stony Brook University, New York, United States of America.
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Definition 1.1. Let V be a vector space. Suppose to any 2 vectors v, u ∈ V there assigned a number from R which will be denoted by 〈v, u〉 such that the following 3 properties hold:
Bilinearity • 〈au 1 + bu 2 , v〉 = a〈u 1 , v〉 + b〈u 2 , v〉
Reflexivity 〈u, v〉 = 〈v, u〉
Positivity 〈u, u〉 ≥ 0 ; moreover if 〈u, u〉 = 0, then u = 0.
Any function which satisfy properties above is called a scalar (inner) product. A vector space V with a scalar product is called a (real) Euclidean space.
Now we will give popular examples of the scalar products in different spaces.
Rn^ The scalar product of 2 vectors x and y from Rn^ can be defined as following: if
x = (x 1 , x 2 ,... , xn) and y = (y 1 , y 2 ,... , yn)
then 〈x, y〉 = x 1 y 2 + x 2 y 2 + · · · + xnyn. If the vectors are represented by column-vectors, i.e. by n × 1-matrices, then
〈x, y〉 = x>y.
Example 1.2. Let x = (1, 2 , 3) and y = (3, − 1 , 4). Then 〈x, y〉 = 1·3+2·(−1)+3·4 = 13.
Actually, there are other ways of defining a scalar product in the vector space Rn. For example, given positive numbers a 1 , a 2 ,... , an we can define the scalar product to be
〈(x 1 , x 2 ,... , xn), (y 1 , y 2 ,... , yn)〉 = a 1 (x 1 y 1 ) + a 2 (x 2 y 2 ) + · · · + an(xnyn).
Mm,n The scalar product of 2 m × n-matrices A and B such that
a 11 a 12... a 1 n a 21 a 22... a 2 n
................... am 1 am 2... amn
and^ B^ =
b 11 b 12... b 1 n b 21 b 22... b 2 n
................... bm 1 bm 2... bmn
is equal to the sum of products of the corresponding entries: 〈A, B〉 = a 11 b 11 + a 12 b 12 + · · · + a 1 nb 1 n
tr
Now, using trace we can write that 〈A, B〉 = tr(AB>). This is true, since diagonal elements of AB>^ are the following: (1, 1) : a 11 b 11 + a 12 b 12 + · · · + a 1 nb 1 n; (2, 2) : a 21 b 21 + a 22 b 22 + · · · + a 2 nb 2 n;
... (m, m) : am 1 bm 1 + am 2 bm 2 + · · · + amnbmn.
Pn(t), P (t), C[0, 1] Here we’re considering the spaces of polynomials and the space of con- tinuous functions on the interval [0, 1]. If f, g are 2 functions (or polynomials) we can define their scalar product by the following formula:
〈f, g〉 =
0
f (t)g(t) dt.
Moreover, we can consider the space of all functions which are continuous on the interval [a, b], and in this case the scalar product will be defined as
〈f, g〉 =
∫ (^) b
a
f (t)g(t) dt.
Positivity ‖v‖ ≥ 0. Moreover, ‖v‖ = 0 if and only if v = 0.
Linearity ‖kv‖ = |k|‖v‖.
Triangle inequality ‖u + v‖ ≤ ‖u‖ + ‖v‖.
Proof. Positivity: Directly follows from the definition of the norm. Linearity: ‖kv‖ =
〈kv, kv〉 =
k^2 〈v, v〉 = |k|
〈v, v〉 = |k|‖v‖. Triangle inequality: Let’s consider ‖u + v‖ ≥ 0. We can rewrite it in the following way:
‖u + v‖^2 = 〈u + v, u + v〉 = 〈u, u〉 + 2〈u, v〉 + 〈v, v〉 = ‖u‖^2 + 2〈u, v〉 + ‖v‖^2 by C-B-S inequality ≤ ‖u‖^2 + 2‖u‖‖v‖ + ‖v‖^2 = (‖u‖ + ‖v‖)^2
Now, taking roots of both sides we get
‖u + v‖ ≤ ‖u‖ + ‖v‖.
Geometrically speaking, the last inequality means that the side of the triangle is less then or equal to the sum of other 2 sides:
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u+v v