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6.1 Inner Products and Norms

Many geometric notions such as angle, length, and perpendicularity in R2 $R^{2}$ and R3 $R^{3}$ may be extended to more general real and complex vector spaces. All of these ideas are related to the concept of inner product.

Definition.

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 and y in V, a scalar in F, denoted ⟨x, y⟩ $〈 x, y 〉$ , such that for all x, y, and z in V and all c in F, the following hold:

(a) ⟨x+z, y⟩=⟨x, y⟩+⟨z, y⟩. $〈 x + z, y 〉 = 〈 x, y 〉 + 〈 z, y 〉 .$
(b) ⟨cx, y⟩=c⟨x, y⟩. $〈 c x, y 〉 = c 〈 x, y 〉 .$
(c) ⟨x, y⟩¯¯¯¯¯¯¯¯¯=⟨y, x⟩ $\bar{〈 x, y 〉} = 〈 y, x 〉$ , where the bar denotes complex conjugation.
(d) If x≠0 $x \neq 0$ then ⟨x, x⟩ $〈 x, x 〉$ is a positive real number.

Note that (c) reduces to ⟨x, y⟩=⟨y, x⟩ $〈 x, y 〉 = 〈 y, x 〉$ if F=R. $F = R .$ Conditions (a) and (b) simply require that the inner product be linear in the first component.

It is easily shown that if a1, a2, …, an∈F $a_{1}, a_{2}, \dots, a_{n} \in F$ and y, v1, v2, …, vn∈V, $y, v_{1}, v_{2}, \dots, v_{n} \in V,$ then

⟨ \sum i = 1 n a i v i, y ⟩ = \sum i = 1 n a i ⟨ v i, y ⟩ .

$⟨ \sum_{i = 1}^{n} a_{i} v_{i}, y ⟩ = \sum_{i = 1}^{n} a_{i} 〈 v_{i}, y 〉 .$

Example 1

For x=(a1, a2, …, an) $x = (a_{1}, a_{2}, \dots, a_{n})$ and y=(b1, b2, …, bn) $y = (b_{1}, b_{2}, \dots, b_{n})$ in Fn $F^{n}$ , define

⟨ x, y ⟩ = \sum i = 1 n a i b ¯ i .

$〈 x, y 〉 = \sum_{i = 1}^{n} a_{i} {\bar{b}}_{i} .$

The verification that ⟨⋅, ⋅⟩ $〈 \cdot, \cdot 〉$ satisfies conditions (a) through (d) is easy. For example, if z=(c1, c2, …, cn) $z = (c_{1}, c_{2}, \dots, c_{n})$ , we have for (a)

⟨ x + z, y ⟩ = = \sum i = 1 n (a i + c i) b ¯ i = \sum i = 1 n a i b ¯ i + \sum i = 1 n c i b ¯ i ⟨ x, y ⟩ + ⟨ z, y ⟩ .

$\begin{array}{rcl} 〈 x + z, y 〉 & = & \sum_{i = 1}^{n} (a_{i} + c_{i}) {\bar{b}}_{i} = \sum_{i = 1}^{n} a_{i} {\bar{b}}_{i} + \sum_{i = 1}^{n} c_{i} {\bar{b}}_{i} \\ = & 〈 x, y 〉 + 〈 z, y 〉 . \end{array}$

Thus, for x=(1+i, 4) $x = (1 + i, 4)$ and y=(2−3i, 4+5i) $y = (2 - 3 i, 4 + 5 i)$ in C2 $C^{2}$ , ⟨x, y⟩=(1+i)(2+3i)+4(4−5i)=15−15i. $〈 x, y 〉 = (1 + i) (2 + 3 i) + 4 (4 - 5 i) = 15 - 15 i .$

The inner product in Example 1 is called the standard inner product on Fn $F^{n}$ . When F=R $F = R$ , the conjugations are not needed, and in early courses this standard inner product is usually called the dot product and is denoted by x⋅y $x \cdot y$ instead of ⟨x, y⟩ $〈 x, y 〉$ .

Example 2

If ⟨x, y⟩ $〈 x, y 〉$ is any inner product on a vector space V and r>0 $r > 0$ , we may define another inner product by the rule ⟨x, y⟩′=r⟨x, y⟩ ${〈 x, y 〉}^{'} = r 〈 x, y 〉$ . If r≤0 $r \leq 0$ , then (d) would not hold.

Example 3

Let V=C([0, 1]) $V = C ([0, 1])$ , the vector space of real-valued continuous functions on [0, l]. For f, g∈V $f, g \in V$ , define ⟨f, g⟩=∫10f(t)g(t) dt $〈 f, g 〉 = \int_{0}^{1} f (t) g (t) d t$ . Since the preceding integral is linear in f, (a) and (b) are immediate, and (c) is trivial. If f≠0 $f \neq 0$ , then f2 $f^{2}$ is bounded away from zero on some subinterval of [0, 1] (continuity is used here), and hence ⟨f, f⟩=∫10[f(t)]2 dt>0 $〈 f, f 〉 = \int_{0}^{1} {[f (t)]}^{2} d t > 0$ .

Definition.

Let A∈Mm×n(F) $A \in M_{m \times n} (F)$ . We define the conjugate transpose or adjoint of A to be the n×m $n \times m$ matrix A* such that (A*)ij=A¯¯¯ji ${(A *)}_{i j} = {\bar{A}}_{j i}$ for all i, j.

Example 4

Let

A = (i 2 1 + 2 i 3 + 4 i) .

$A = (\begin{array}{r} i & 1 + 2 i \\ 2 & 3 + 4 i \end{array}) .$

Then

A * = (- i 1 - 2 i 2 3 - 4 i) .

$A * = (\begin{array}{c} - i & 2 \\ 1 - 2 i & 3 - 4 i \end{array}) .$

Notice that if x and y are viewed as column vectors in Fn $F^{n}$ , then ⟨x, y⟩=y*x. $〈 x, y 〉 = y * x .$

The conjugate transpose of a matrix plays a very important role in the remainder of this chapter. In the case that A has real entries, A* is simply the transpose of A.

Example 5

Let V=Mn×n(F) $V = M_{n \times n} (F)$ , and define ⟨A, B⟩=tr(B*A) $〈 A, B 〉 = tr (B * A)$ for A, B∈V $A, B \in V$ . (Recall that the trace of a matrix A is defined by tr(A)=∑ni=1Aii $tr (A) = \sum_{i = 1}^{n} A_{i i}$ .) We verify that (a) and (d) of the definition of inner product hold and leave (b) and (c) to the reader. For this purpose, let A, B, C∈V $A, B, C \in V$ . Then (using Exercise 6 of Section 1.3)

⟨ A + B, C ⟩ = = tr (C * (A + B)) = tr (C * A + C * B) tr (C * A) + tr (C * B) = ⟨ A, C ⟩ + ⟨ B, C ⟩ .

$\begin{array}{rcl} 〈 A + B, C 〉 & = & tr (C * (A + B)) = tr (C * A + C * B) \\ = & tr (C * A) + tr (C * B) = 〈 A, C 〉 + 〈 B, C 〉 . \end{array}$

Also

⟨ A, A ⟩ = = tr (A * A) = \sum i = 1 n (A * A) i i = \sum i = 1 n \sum k = 1 n (A *) i k A k i \sum i = 1 n \sum k = 1 n A ¯ ¯ ¯ k i A k i = \sum i = 1 n \sum k = 1 n | A k i | 2 .

$\begin{array}{rcl} 〈 A, A 〉 & = & tr (A * A) = \sum_{i = 1}^{n} {(A * A)}_{i i} = \sum_{i = 1}^{n} \sum_{k = 1}^{n} {(A *)}_{i k} A_{k i} \\ = & \sum_{i = 1}^{n} \sum_{k = 1}^{n} {\bar{A}}_{k i} A_{k i} = \sum_{i = 1}^{n} \sum_{k = 1}^{n} | A_{k i} |^{2} . \end{array}$

Now if A≠O $A \neq O$ , then Aki≠0 $A_{k i} \neq 0$ for some k and i. So ⟨A, A⟩>0 $〈 A, A 〉 > 0$ .

The inner product on Mn×n(F) $M_{n \times n} (F)$ in Example 5 is called the Frobenius inner product.

A vector space V over F endowed with a specific inner product is called an inner product space. If F=C $F = C$ , we call V a complex inner product space, whereas if F=R $F = R$ , we call V a real inner product space.

It is clear that if V has an inner product ⟨x, y⟩ $〈 x, y 〉$ and W is a subspace of V, then W is also an inner product space when the same function ⟨x, y⟩ $〈 x, y 〉$ is restricted to the vectors x, y∈W $x, y \in W$ .

Thus Examples 1, 3, and 5 also provide examples of inner product spaces. For the remainder of this chapter, Fn $F^{n}$ denotes the inner product space with the standard inner product as defined in Example 1. Likewise, Mn×n(F) $M_{n \times n} (F)$ denotes the inner product space with the Frobenius inner product as defined in Example 5. The reader is cautioned that two distinct inner products on a given vector space yield two distinct inner product spaces. For instance, it can be shown that both

⟨ f (x), g (x) ⟩ 1 = \int 10 f (t) g (t) d t and ⟨ f (x), g (x) ⟩ 2 = \int 1 - 1 f (t) g (t) d t

${〈 f (x), g (x) 〉}_{1} = \int_{0}^{1} f (t) g (t) d t and {〈 f (x), g (x) 〉}_{2} = \int_{- 1}^{1} f (t) g (t) d t$

are inner products on the vector space P(R). Even though the underlying vector space is the same, however, these two inner products yield two different inner product spaces. For example, the polynomials f(x)=x $f (x) = x$ and g(x)=x2 $g (x) = x^{2}$ are orthogonal in the second inner product space, but not in the first.

A very important inner product space that resembles C([0, 1]) is the space H of continuous complex-valued functions defined on the interval [0, 2π] $[0, 2 π]$ with the inner product

⟨ f, g ⟩ = 1 2 π \int 2 π 0 f (t) g (t) ¯ ¯ ¯ ¯ ¯ ¯ d t .

$〈 f, g 〉 = \frac{1}{2 π} \int_{0}^{2 π} f (t) \bar{g (t)} d t .$

The reason for the constant 1/2π $1 / 2 π$ will become evident later. This inner product space, which arises often in the context of physical situations, is examined more closely in later sections.

At this point, we mention a few facts about integration of complex-valued functions. First, the imaginary number i can be treated as a constant under the integration sign. Second, every complex-valued function f may be written as f=f1+if2 $f = f_{1} + i f_{2}$ , where f1 $f_{1}$ and f2 $f_{2}$ are real-valued functions. Thus we have

\int f d t = \int f 1 d t + i \int f 2 d t and \int f ¯ ¯ ¯ ¯ ¯ ¯ d t = \int f ¯ d t .

$\int f d t = \int f_{1} d t + i \int f_{2} d t and \bar{\int f} d t = \int \bar{f} d t .$

From these properties, as well as the assumption of continuity, it follows that H is an inner product space (see Exercise 16(a)).

Some properties that follow easily from the definition of an inner product are contained in the next theorem.

Theorem 6.1.

Let V be an inner product space. Then for x, y, z∈V $x, y, z \in V$ and c∈F $c \in F$ , the following statements are true.

(a) ⟨x, y+z⟩=⟨x, y⟩+⟨x, z⟩. $〈 x, y + z 〉 = 〈 x, y 〉 + 〈 x, z 〉 .$
(b) ⟨x, cy⟩=c¯⟨x, y⟩. $〈 x, c y 〉 = \bar{c} 〈 x, y 〉 .$
(c) ⟨x, 0⟩=⟨0, x⟩=0. $〈 x, 0 〉 = 〈 0, x 〉 = 0.$
(d) ⟨x, x⟩=0 $〈 x, x 〉 = 0$ if and only if x=0 $x = 0$ .
(e) if ⟨x, y⟩=⟨x, z⟩ $〈 x, y 〉 = 〈 x, z 〉$ for all x∈V $x \in V$ then y=z $y = z$ .

Proof.

(a) We have

⟨ x, y + z ⟩ = = ⟨ y + z, x ⟩ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = ⟨ y, x ⟩ + ⟨ z, x ⟩ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ⟨ y, x ⟩ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ + ⟨ z, x ⟩ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = ⟨ x, y ⟩ + ⟨ x, z ⟩ .

$\begin{array}{rcl} 〈 x, y + z 〉 & = & \bar{〈 y + z, x 〉} = \bar{〈 y, x 〉 + 〈 z, x 〉} \\ = & \bar{〈 y, x 〉} + \bar{〈 z, x 〉} = 〈 x, y 〉 + 〈 x, z 〉 . \end{array}$

The proofs of (b), (c), (d), and (e) are left as exercises.

The reader should observe that (a) and (b) of Theorem 6.1 show that the inner product is conjugate linear in the second component.

In order to generalize the notion of length in R3 $R^{3}$ to arbitrary inner product spaces, we need only observe that the length of x=(a, b, c)∈R3 $x = (a, b, c) \in R^{3}$ is given by a2+b2+c2−−−−−−−−−−√=⟨x, x⟩−−−−−√ $\sqrt{a^{2} + b^{2} + c^{2}} = \sqrt{〈 x, x 〉}$ . This leads to the following definition.

Definition.

Let V be an inner product space. For x∈V $x \in V$ , we define the norm or length of x by ||x||=⟨x, x⟩−−−−−√ $| | x | | = \sqrt{〈 x, x 〉}$ .

Example 6

Let V=Fn $V = F^{n}$ . If x=(a1, a2 …, an) $x = (a_{1}, a_{2} \dots, a_{n})$ , then

| | x | | = | | (a 1, a 2 \dots, a n) | | = [\sum i = 1 n | a i | 2] 1 / 2

$| | x | | = | | (a_{1}, a_{2} \dots, a_{n}) | | = {[\sum_{i = 1}^{n} | a_{i} |^{2}]}^{1 / 2}$

is the Euclidean definition of length. Note that if n=1 $n = 1$ , we have ||a||=|a| $| | a | | = | a |$ .

As we might expect, the well-known properties of Euclidean length in R3 $R^{3}$ hold in general, as shown next.

Theorem 6.2.

Let V be an inner product space over F. Then for all x, y∈V $x, y \in V$ and c∈F $c \in F$ the following statements are true.

(a) ||cx||=|c|⋅||x||. $| | c x | | = | c | \cdot | | x | | .$
(b) ||x||=0 $| | x | | = 0$ if and only if x=0 $x = 0$ . In any case, ||x||≥0 $| | x | | \geq 0$ .
(c) (Cauchy-Schwarz Inequality) |⟨x, y⟩|≤||x||⋅||y|| $| 〈 x, y 〉 | \leq | | x | | \cdot | | y | |$ .
(d) (Triangle Inequality) ||x+y||≤||x||+||y|| $| | x + y | | \leq | | x | | + | | y | |$ .

Proof.

We leave the proofs of (a) and (b) as exercises.

0 \leq ⟨ x - c y, x - c y ⟩ = = ⟨ x, x - c y ⟩ - c ⟨ y, x - c y ⟩ ⟨ x, x ⟩ - c ¯ ⟨ x, y ⟩ - c ⟨ y, x ⟩ + c c ¯ ⟨ y, y ⟩ .

$\begin{array}{rcl} 0 \leq 〈 x - c y, x - c y 〉 & = & 〈 x, x - c y 〉 - c 〈 y, x - c y 〉 \\ = & 〈 x, x 〉 - \bar{c} 〈 x, y 〉 - c 〈 y, x 〉 + c \bar{c} 〈 y, y 〉 . \end{array}$

In particular, if we set

c = ⟨ x , y ⟩ ⟨ y , y ⟩,

$c = \frac{〈 x, y 〉}{〈 y, y 〉},$

then each of c¯⟨x, y⟩, c⟨y, x⟩ $\bar{c} 〈 x, y 〉, c 〈 y, x 〉$ , and cc¯⟨y, y⟩ $c \bar{c} 〈 y, y 〉$ equals ⟨x, y⟩⟨y, x⟩⟨y, y⟩=|⟨x, y⟩|2||y||2 $\frac{〈 x, y 〉 〈 y, x 〉}{〈 y, y 〉} = \frac{| 〈 x, y 〉 |^{2}}{| | y | |^{2}}$ . So the preceding inequality becomes

0 \leq ⟨ x, x ⟩ - | ⟨ x , y ⟩ | 2 ⟨ y , y ⟩ = | | x | | 2 - | ⟨ x , y ⟩ | 2 | | y | | 2,

$0 \leq 〈 x, x 〉 - \frac{| 〈 x, y 〉 |^{2}}{〈 y, y 〉} = | | x | |^{2} - \frac{| 〈 x, y 〉 |^{2}}{| | y | |^{2}},$

from which (c) follows.

(d) We have

$\begin{array}{rcl} | | x + y | |^{2} & = & 〈 x + y, x + y 〉 = 〈 x, x 〉 + 〈 y, x 〉 + 〈 x, y 〉 + 〈 y, y 〉 \\ = & | | x | |^{2} + 2 ℜ 〈 x, y 〉 + | | y | |^{2} \\ \leq & | | x | |^{2} + 2 | 〈 x, y 〉 | + | | y | |^{2} \\ \leq & | | x | |^{2} + 2 | | x | | \cdot | | y | | + | | y | |^{2} \\ = & {(| | x | | + | | y | |)}^{2}, \end{array}$

where $ℜ 〈 x, y 〉$ denotes the real part of the complex number $〈 x, y 〉$ . Note that we used (c) to prove (d).

The case when equality results in (c) and (d) is considered in Exercise 15.

Example 7

For $F^{n}$ , we may apply (c) and (d) of Theorem 6.2 to the standard inner product to obtain the following well-known inequalities:

$| \sum_{i = 1}^{n} a_{i} {\bar{b}}_{i} | \leq {[\sum_{i = 1}^{n} | a_{i} |^{2}]}^{1 / 2} {[\sum_{i = 1}^{n} | b_{i} |^{2}]}^{1 / 2}$

and

${[\sum_{i = 1}^{n} | a_{i} + b_{i} |^{2}]}^{1 / 2} \leq {[\sum_{i = 1}^{n} | a_{i} |^{2}]}^{1 / 2} + {[\sum_{i = 1}^{n} | b_{i} |^{2}]}^{1 / 2} .$

The reader may recall from earlier courses that, for x and y in $R^{3}$ or $R^{2}$ , we have that $〈 x, y 〉 = | | x | | \cdot | | y | | \cos θ$ , where $θ (0 \leq θ \leq π)$ denotes the angle between x and y. This equation implies (c) immediately since $| \cos θ | \leq 1$ . Notice also that nonzero vectors x and y are perpendicular if and only if $\cos θ = 0$ , that is, if and only if $〈 x, y 〉 = 0$ .

We are now at the point where we can generalize the notion of perpendicularity to arbitrary inner product spaces.

Definitions.

Let V be an inner product space. Vectors x and y in V are orthogonal (or perpendicular) if $〈 x, y 〉 = 0$ . A subset S of V is orthogonal if any two distinct vectors in S are orthogonal. A vector x in V is a unit vector if $| | x | | = 1$ . Finally, a subset S of V is orthonormal if S is orthogonal and consists entirely of unit vectors.

Note that if $S = {v_{1}, v_{2}, \dots}$ , then S is orthonormal if and only if $〈 v_{i}, v_{j} 〉 = δ_{i j}$ , where $δ_{i j}$ denotes the Kronecker delta. Also, observe that multiplying vectors by nonzero scalars does not affect their orthogonality and that if x is any nonzero vector, then $(1 / | | x | |) x$ is a unit vector. The process of multiplying a nonzero vector by the reciprocal of its length is called normalizing.

Example 8

In $F^{3}, {(1, 1, 0), (1, - 1, 1), (- 1, 1, 2)}$ is an orthogonal set of nonzero vectors, but it is not orthonormal; however, if we normalize the vectors in the set, we obtain the orthonormal set

${\frac{1}{\sqrt{2}} (1, 1, 0), \frac{1}{\sqrt{3}} (1, - 1, 1), \frac{1}{\sqrt{6}} (- 1, 1, 2)} .$

Our next example is of an infinite orthonormal set that is important in analysis. This set is used in later examples in this chapter.

Example 9

Recall the inner product space H (defined on page 330). We introduce an important orthonormal subset S of H. For what follows, i is the imaginary number such that $i^{2} = - 1$ . For any integer n, let $f_{n} (t) = e^{i n t}$ , where $0 \leq t \leq 2 π$ . (Recall that $e^{i n t} = \cos n t + i \sin n t$ .) Now define $S = {f_{n} : n is an integer}$ .

Clearly S is a subset of H. Using the property that $\bar{e^{i t}} = e^{- i t}$ for every real number t, we have, for $m \neq n$ ,

$\begin{array}{rcl} 〈 f_{m}, f_{n} 〉 & = & \frac{1}{2 π} \int_{0}^{2 π} e^{i m t} \bar{e^{i n t}} d t = \frac{1}{2 π} \int_{0}^{2 π} e^{i (m - n) t} d t \\ = & \frac{1}{2 π i (m - n)} e^{i (m - n) t} |_{0}^{2 π} = 0. \end{array}$

Also,

$〈 f_{n}, f_{n} 〉 = \frac{1}{2 π} \int_{0}^{2 π} e^{i (n - n) t} d t = \frac{1}{2 π} \int_{0}^{2 π} 1 d t = 1 .$

In other words, $〈 f_{m}, f_{n} 〉 = δ_{m n}$ .

Exercises

Label the following statements as true or false.
1. (a) An inner product is a scalar-valued function on the set of ordered pairs of vectors.
2. (b) An inner product space must be over the field of real or complex numbers.
3. (c) An inner product is linear in both components.
4. (d) There is exactly one inner product on the vector space $R^{n}$ .
5. (e) The triangle inequality only holds in finite-dimensional inner product spaces.
6. (f) Only square matrices have a conjugate-transpose.
7. (g) If x, y, and z are vectors in an inner product space such that $〈 x, y 〉 = 〈 x, z 〉$ , then $y = z$ .
8. (h) If $〈 x, y 〉 = 0$ for all x in an inner product space, then $y = 0$ .
Let $x = (2, 1 + i, i)$ and $y = (2 - i, 2, 1 + 2 i)$ be vectors in $C^{3}$ . Compute $〈 x, y 〉, | | x | |, | | y | |$ , and $| | x + y | |$ . Then verify both the Cauchy-Schwarz inequality and the triangle inequality.
In C([0, 1]), let $f (t) = t$ and $g (t) = e^{t}$ . Compute $〈 f, g 〉$ (as defined in Example 3), $| | f | |, | | g | |,$ and $| | f + g | |$ . Then verify both the Cauchy-Schwarz inequality and the triangle inequality.
1. (a) Complete the proof in Example 5 that $〈 \cdot, \cdot 〉$ is an inner product (the Frobenius inner product) on $M_{n \times n} (F)$ .
2. (b) Use the Frobenius inner product to compute $| | A | |, | | B | |$ , and $〈 A, B 〉$ for
  
  $A = (\begin{array}{c} 1 & 2 + i \\ 3 & i \end{array}) and B = (\begin{array}{c} 1 + i & 0 \\ i & - i \end{array}) .$
In $C^{2}$ , show that $〈 x, y 〉 = x A y *$ is an inner product, where

$A = (\begin{array}{r} 1 & i \\ - i & 2 \end{array}) .$

Compute $〈 x, y 〉$ for $x = (1 - i, 2 + 3 i)$ and $y = (2 + i, 3 - 2 i)$ .
Complete the proof of Theorem 6.1.
Complete the proof of Theorem 6.2.
Provide reasons why each of the following is not an inner product on the given vector spaces.
1. (a) $〈 (a, b), (c, d) 〉 = a c - b d$ on $R^{2}$
2. (b) $〈 A, B 〉 = tr (A + B)$ on $M_{2 \times 2} (R)$
3. (c) $〈 f (x), g (x) 〉 = \int_{0}^{1} f^{'} (t) g (t) d t$ on P(R), where ‘ denotes differentiation.
Let $β$ be a basis for a finite-dimensional inner product space.
1. (a) Prove that if $〈 x, z 〉 = 0$ for all $z \in β$ , then $x = 0$ .
2. (b) Prove that if $〈 x, z 〉 = 〈 y, z 〉$ for all $z \in β$ , then $x = y$ .
† Let V be an inner product space, and suppose that x and y are orthogonal vectors in V. Prove that $| | x + y | |^{2} = | | x | |^{2} + | | y | |^{2}$ . Deduce the Pythagorean theorem in $R^{2}$ . Visit goo.gl/1iTZzC for a solution.
Prove the parallelogram law on an inner product space V; that is, show that

$| | x + y | |^{2} + | | x - y | |^{2} = 2 | | x | |^{2} + 2 | | y | |^{2} for all x, y \in V .$

What does this equation state about parallelograms in $R^{2}$ ?
† Let ${v_{1}, v_{2}, \dots, v_{k}}$ be an orthogonal set in V, and let $a_{1}, a_{2}, \dots, a_{k}$ be scalars. Prove that

${‖ \sum_{i = 1}^{k} a_{i} v_{i} ‖}^{2} = \sum_{i = 1}^{k} | a_{i} |^{2} | | v_{i} | |^{2} .$
Suppose that ${〈 \cdot, \cdot 〉}_{1}$ and ${〈 \cdot, \cdot 〉}_{2}$ are two inner products on a vector space V. Prove that $〈 \cdot, \cdot 〉 = {〈 \cdot, \cdot 〉}_{1} + {〈 \cdot, \cdot 〉}_{2}$ is another inner product on V.
Let A and B be $n \times n$ matrices, and let c be a scalar. Prove that $(A + c B) * = A * + \bar{c} B *$ .
1. (a) Prove that if V is an inner product space, then $| 〈 x, y 〉 | = | | x | | \cdot | | y | |$ if and only if one of the vectors x or y is a multiple of the other. Hint: If the identity holds and $y \neq 0$ , let
  
  $a = \frac{〈 x, y 〉}{| | y | |^{2}},$
  
  and let $z = x - a y$ . Prove that y and z are orthogonal and
  
  $| a | = \frac{| | x | |}{| | y | |} .$
  
  Then apply Exercise 10 to $| | x | |^{2} = | | a y + z | |^{2}$ to obtain $| | z | | = 0$ .
2. (b) Derive a similar result for the equality $| | x + y | | = | | x | | + | | y | |$ , and generalize it to the case of n vectors.
1. (a) Show that the vector space H with $〈 \cdot, \cdot 〉$ defined on page 330 is an inner product space.
2. (b) Let $V = C ([0, 1])$ , and define
  
  $〈 f, g 〉 = \int_{0}^{1 / 2} f (t) g (t) d t .$
  
  Is this an inner product on V?
Let T be a linear operator on an inner product space V, and suppose that $‖ T (x) ‖ = ‖ x ‖$ for all x. Prove that T is one-to-one.
Let V be a vector space over F, where $F = R$ or $F = C$ , and let W be an inner product space over F with inner product $〈 \cdot, \cdot 〉$ . If $T : V \to W$ is linear, prove that ${〈 x, y 〉}^{'} = 〈 T (x), T (y) 〉$ defines an inner product on V if and only if T is one-to-one.
Let V be an inner product space. Prove that
1. (a) $| | x \pm y | |^{2} = | | x | |^{2} \pm 2 ℜ 〈 x, y 〉 + | | y | |^{2}$ for all $x, y \in V$ , where $ℜ 〈 x, y 〉$ denotes the real part of the complex number $〈 x, y 〉$ .
2. (b) $| | | x | | - | | y | | | \leq | | x - y | |$ for all $x, y \in V$ .
Let V be an inner product space over F. Prove the polar identities: For all $x, y \in V$ ,
1. (a) $〈 x, y 〉 = \frac{1}{4} | | x + y | |^{2} - \frac{1}{4} | | x - y | |^{2}$ if $F = R;$
2. (b) $〈 x, y 〉 = \frac{1}{4} \sum_{k = 1}^{4} i^{k} | | x + i^{k} y | |^{2}$ if $F = C$ , where $i^{2} = - 1$ .
Let A be an $n \times n$ matrix. Define

$A_{1} = \frac{1}{2} (A + A *) and A_{2} = \frac{1}{2 i} (A - A *) .$
1. (a) Prove that $A_{1}^{*} = A_{1}, A_{2}^{*} = A_{2,}$ and $A = A_{1} + i A_{2}$ . Would it be reasonable to define $A_{1}$ and $A_{2}$ to be the real and imaginary parts, respectively, of the matrix A?
2. (b) Let A be an $n \times n$ matrix. Prove that the representation in (a) is unique. That is, prove that if $A = B_{1} + i B_{2}$ , where $B_{1}^{*} = B_{1}$ and $B_{2}^{*} = B_{2}$ , then $B_{1} = A_{1}$ and $B_{2} = A_{2}$ .
Let V be a real or complex vector space (possibly infinite-dimensional), and let $β$ be a basis for V. For $x, y \in V$ there exist $v_{1}, v_{2}, \dots, v_{n} \in β$ such that

$x = \sum_{i = 1}^{n} a_{i} v_{i} and y = \sum_{i = 1}^{n} b_{i} v_{i} .$

Define

$〈 x, y 〉 = \sum_{i = 1}^{n} a_{i} {\bar{b}}_{i} .$
1. (a) Prove that $〈 \cdot, \cdot 〉$ is an inner product on V and that $β$ is an orthonormal basis for V. Thus every real or complex vector space may be regarded as an inner product space.
2. (b) Prove that if $V = R^{n}$ or $V = C^{n}$ and $β$ is the standard ordered basis, then the inner product defined above is the standard inner product.
Let $V = F^{n}$ , and let $A \in M_{n \times n} (F)$ .
1. (a) Prove that $〈 x, A y 〉 = 〈 A * x, y 〉$ for all $x, y \in V$ .
2. (b) Suppose that for some $B \in M_{n \times n} (F)$ , we have $〈 x, A y 〉 = 〈 B x, y 〉$ for all $x, y \in V$ . Prove that $B = A *$ .
3. (c) Let $α$ be the standard ordered basis for V. For any orthonormal basis $β$ for V, let Q be the $n \times n$ matrix whose columns are the vectors in $β$ . Prove that $Q^{*} = Q^{- 1}$ .
4. (d) Define linear operators T and U on V by $T (x) = A x$ and $U (x) = A * x$ . Show that ${[U]}_{β} = {[T]}_{β}^{*}$ for any orthonormal basis $β$ for V.
Let V be a complex inner product space with an inner product $〈 \cdot, \cdot 〉$ . Let $[\cdot, \cdot]$ be the real-valued function such that [x, y] is the real part of the complex number $〈 x, y 〉$ for all $x, y \in V$ . Prove that $[\cdot, \cdot]$ is an inner product for V, where V is regarded as a vector space over R. Prove, furthermore, that $[x, i x] = 0$ for all $x \in V$ .
Let V be a vector space over C, and suppose that $[\cdot, \cdot]$ is a real inner product on V, where V is regarded as a vector space over R, such that $[x, i x] = 0$ for all $x \in V$ . Let $〈 \cdot, \cdot 〉$ be the complex-valued function defined by

$〈 x, y 〉 = [x, y] + i [x, i y] for x, y \in V .$

Prove that $〈 \cdot, \cdot 〉$ is a complex inner product on V.

The following definition is used in Exercises 26–30.

Definition.

Let V be a vector space over F, where F is either R or C. Regardless of whether V is or is not an inner product space, we may still define a norm $| | \cdot | |_{v}$ as a real-valued function on V satisfying the following three conditions for all $x, y \in V$ and $a \in F$ :

(1) $| | x | |_{v} \geq 0,$ and $| | x | |_{v} = 0$ if and only if $x = 0$ .
(2) $| | a x | |_{v} = | a | \cdot | | x | |_{v} .$
(3) $| | x + y | |_{v} \leq | | x | |_{v} + | | y | |_{v} .$

Prove that the following are norms on the given vector spaces V.
1. (a) $V = R^{2}; | | (a, b) | |_{v} = | a | + | b |$ for all $(a, b) \in V$
2. (b) $V = C ([0, 1]); | | f | |_{v} = \max_{t \in [0, 1]} | f (t) |$ for all $f \in V$
3. (c) $V = C ([0, 1]); | | f | |_{v} = \int_{0}^{1} | f (t) | d t$ for all $f \in V$
4. (d) $V = M_{m \times n} (F); | | A | |_{v} = \max_{i, j} | A_{i j} |$ for all $A \in V$
Use Exercise 11 to show that there is no inner product $〈 \cdot, \cdot 〉$ on $R^{2}$ such that $| | x | |_{V}^{2} = 〈 x, x 〉$ for all $x \in R^{2}$ if the norm is defined as in Exercise 26(a).
Let $| | \cdot | |_{v}$ be a norm on a vector space V, and define, for each ordered pair of vectors, the scalar $d (x, y) = | | x - y | |_{v}$ , called the distance between x and y. Prove the following results for all $x, y, z \in V$ .
1. (a) $d (x, y) \geq 0.$
2. (b) $d (x, y) = d (y, x) .$
3. (c) $d (x, y) \leq d (x, z) + d (z, y) .$
4. (d) $d (x, x) = 0$ if and only if $x = 0.$
5. (e) $d (x, y) \neq 0$ if $x \neq y .$
Let $| | \cdot | |_{v}$ be a norm on a real vector space V satisfying the parallelogram law given in Exercise 11. Define

$〈 x, y 〉 = \frac{1}{4} [| | x + y | |_{v}^{2} - | | x - y | |_{v}^{2}] .$

Prove that $〈 \cdot, \cdot 〉$ defines an inner product on V such that $| | x | |_{v}^{2} = 〈 x, x 〉$ for all $x \in V$ . Hints:
1. (a) Prove $〈 x, 2 y 〉 = 2 〈 x, y 〉$ for all $x, y \in V$ .
2. (b) Prove $〈 x + u, y 〉 = 〈 x, y 〉 + 〈 u, y 〉$ for all $x, u, y \in V$ .
3. (c) Prove $〈 n x, y 〉 = n 〈 x, y 〉$ for every positive integer n and every $x, y \in V$ .
4. (d) Prove $m 〈 \frac{1}{m} x, y 〉 = 〈 x, y 〉$ for every positive integer m and every $x, y \in V$ .
5. (e) Prove $〈 r x, y 〉 = r 〈 x, y 〉$ for every rational number r and every $x, y \in V$ .
6. (f) Prove $| 〈 x, y 〉 | \leq | | x | |_{v} | | y | |_{v}$ for every $x, y \in V$ . Hint: Condition (3) in the definition of norm can be helpful.
7. (g) Prove that for every $c \in R$ , every rational number r, and every $x, y \in V$ ,
  
  $\begin{array}{l} | c 〈 x, y 〉 - 〈 c x, y 〉 | = | (c - r) 〈 x, y 〉 - 〈 (c - r) x, y 〉 | \\ \leq 2 | c - r | | | x | |_{v} | | y | |_{v} . \end{array}$
8. (h) Use the fact that for any $c \in R, | c - r |$ can be made arbitrarily small, where r varies over the set of rational numbers, to establish item (b) of the definition of inner product.
Let $| | \cdot | |_{v}$ be a norm (as defined on page 337) on a complex vector space V satisfying the parallelogram law given in Exercise 11. Prove that there is an inner product $〈 \cdot, \cdot 〉$ on V such that $| | x | |_{v}^{2} = 〈 x, x 〉$ for all $x \in V$ . Hint: Apply Exercise 29 to V regarded as a vector space over R. Then apply Exercise 25.

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Table of Contents for 6.1 Inner Products and Norms

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Table of Contents for
6.1 Inner Products and Norms