Inner Products and Norms: Definition, Theorem

INNER PRODUCTS AND NORMS

Many geometric notations such as angle, length and perpendicularity in R² and R³ may be extended to real and complex vector spaces. All of these concepts are related to inner product.

Definition: Inner product

Let V be a vector space over F. An inner product on V is a function that assigns to every ordered pair of vectors x & y in V, a scalar in F, denoted <x, y> such that for all x, y, z in V, & all c in F, the following conditions hold.

1. <x+z,y> = <x, y> + <z, y>

2. <cx, y> = c<x, y>

3.  = <y, x>, where the bar denotes complex conjugate.

4. <x, x> > 0 if x≠0.

Note

(i) For Real numbers (ie F=R), <x,y> = <y, x>

(ii) <  a_iv_i,y > =  a_i<v_i,y> where a_i, ∈ F.

Standard inner product of Fⁿ

For x = (a₁, a₂, ... a_n) and y = (b₁, b₂, ... b_n) in Fⁿ.

Let us define (x, y) =  a_ii = α₁₁ + α₂₂ + ... + α_nn.

This standard inner product is usually called the dot product and is denoted by x.y instead of <xy>.

Definition: Norm

Let V be an inner product space. For x ∈ V we define the norm or length of x by || x || = √[<x, x>].

Definition

Let A ∈ M_m×n(F). We define the conjugate transpose or adjoint of A to be the n×m matrix A* such that (A*)_ij =  for all i, j.

Definition

A vector space V over F endowed with a specific inner product is called an inner product space. If F=C, it is called as V a complex inner product space. Whereas if F=R it is called as V a real inner product space.

If V has an inner product <x, y> and W is a sub space of V, then W is also an inner product space when the same function <x, y> is restricted to the vectors x, y ∈ W.

Definition: Dot product

Let U (u₁, u₂, … u_n) & V = (v₁, v₂, …. v_n) then the dot product of UV is defined as U.V.

  U.V = u₁v₂ + v₂.v₂ + …. + u_nv_n

U.V=0 ⇒ U & V are perpendicular vectors.

Theorem 1

Let V be a inner product space, then for x,y,z ∈ V & c ∈ F, the following statements are true:

(a) <x,y+z> = <x, y> + <x, z>

(b) <x, cy> = <x,y>

(c) <x, 0> = <0,x> = 0

(d) <x, x> = 0 if and only if x = 0

(e) If <x,y> = <x, z> for all x ∈ V then y = z

Proof:

(d) <x, x> = 0 iffx=0

Let x=0, then <x, x> = <0,0>>  = 0

By condition <x, x> > 0 if x ≠ 0

Obviously <x, x> = 0 iff x=0

(e) Assume <x, y> = <x, z>, for all x ∈ V

Consider <x,y−z> = <x,y> − <x, z>

= 0 for all x ∈ V

Take x=y−z; <y−z,y−z> = 0

 y−z=0 ⇒ y=z

If x ≠ y−z; then either x=0 (or) y‒z=0

 y = z.

Theorem 2

Let V be an inner product space over F then for all x,y ∈ V & c ∈ F the following statements are true.

(a) ||cx|| = |c|  ||x||

(b) || x || = 0 if and only if x = 0

(c) | <x,y> | ≤ ||x|| ||y || (Cauchy − Schwarz inequality)

(d) || x+y|| ≤ ||x|| + || y || (Triangle inequality)

Proof

(a) || cx || = √<cx, cx>

|| cx ||² = < cx, cx >

= c<x,x>

|| cx ||² = |c|² ||x||²

|| cx || = | c | || x ||

(b) || x || = 0 if and only if x=0

 || x || =0

√[<x, x>] =0

 <x, x> = 0

⇒ x=0

since <x, x> > 0 if x ≠ 0.

(c) | <x, y> |  ≤  ||x|| ||y||

If y=0 then ||y||=0 & <x, y> = <x, 0> = 0.

The inequality holds true.

Assume that y≠0 and for any c ∈ F, then || x − cy ||² ≥ 0

0 ≤ || x−cy ||²

≤ <x−cy, x−cy>

=<x, x−cy> + <−cy, x−cy>

= <x,−cy> + <x, x> + <−cy, x> + <cy, cy>

= <x,x> −<x,y> − c<y, x> + <y,y>

 0 ≤ ||x||² ||y||² − | < x, y> |²

 | <x, y> |² ≤ ||x||² ||y||²

                  ( | <x, y> | ≤  ||x|| ||y|| )

(d) || x + y ||  ≤  ||x|| + ||y||

 || x + y ||² = <x + y, x + y> = <x, x + y> + <y, x+y>

 = <x,x> + <x, y> + <y, x> + <y,y>

= || x ||² + || y ||² + < x, y> + <>

= ||x||² + ||y||² + 2Re<x, y>

 ≤ ||x||² + ||y||² + 2||x|| ||y||

 ||x+y||² = ([[ x || + || y ||)²

 ||x+y|| ≤  [[ x || + || y ||

Definition:

Let V = Fⁿ. If x = (a₁, a₂, a₃, ... a_n) then

 || x || = || (a₁, a₂, a₃, a₄, ... a_n) || =  is the Euclidean definition of length. If n=1 then we have || a || = |a|.