Math Problems Vault

Problem 8A.1

Let $V$ denote the vector space of bounded continuous functions from $\mathbb {R}$ to $\mathbf{F}$ . Let $r_{1}, r_{2}, \ldots$ be a list of the rational numbers. For $f, g \in V$ , define

\left\langle f, g \right\rangle = \sum _{k = 1}^{\infty } \frac{f\left(r_{k}\right) \overline{g\left(r_{k}\right)}}{2^{k}}

Show that $\left\langle \cdot , \cdot \right\rangle$ is an inner product on $V$ .

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Problem 8A.2

Prove that if $\mu$ is a measure and $f, g \in L^{2}(\mu )$ , then

\left\| f\right\| ^{2}\left\| g\right\| ^{2}-\left| \left\langle f, g \right\rangle \right|^{2} = \frac{1}{2} \iint \left|f(x) g(y)-g(x) f(y)\right|^{2} d \mu (y) d \mu (x)

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Problem 8A.3

(a)(b)

Suppose $f$ and $g$ are elements of an inner product space and

\left\| f+g\right\| ^{2} = \left\| f\right\| ^{2}+\left\| g\right\| ^{2}

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(a)

Prove that if $\mathbf{F} = \mathbb {R}$ , then $f$ and $g$ are orthogonal.

(b)

Give an example to show that if $\mathbf{F} = \mathbb {C}$ , then $f$ and $g$ can satisfy the equation above without being orthogonal.

Problem 8A.4

Find $a, b \in \mathbb {R}^{3}$ such that $a$ is a scalar multiple of $(1,6,3), b$ is orthogonal to $(1,6,3)$ , and $(5,4,-2) = a+b$ .

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Problem 8A.5

Prove that

16 \leq (a+b+c+d)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)

for all positive numbers $a, b, c, d$ , with equality if and only if $a = b = c = d$ .

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Problem 8A.6

Prove that the square of the average of each finite list of real numbers containing at least two distinct real numbers is less than the average of the squares of the numbers in that list.

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Problem 8A.7

Suppose $f$ and $g$ are elements of an inner product space and $\left\| f\right\| \leq 1$ and $\left\| g\right\| \leq 1$ . Prove that

\sqrt{1-\left\| f\right\| ^{2}} \sqrt{1-\left\| g\right\| ^{2}} \leq 1-\left| \left\langle f, g \right\rangle \right|.

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Problem 8A.8

Suppose $a$ and $b$ are nonzero elements of $\mathbb {R}^{2}$ . Prove that

\left\langle a, b \right\rangle = \left\| a\right\| \left\| b\right\| \cos \theta

where $\theta$ is the angle between $a$ and $b$ (thinking of $a$ as the vector whose initial point is the origin and whose end point is $a$ , and similarly for $b$ ).

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Problem 8A.9

The angle between two vectors (thought of as arrows with initial point at the origin) in $\mathbb {R}^{2}$ or $\mathbb {R}^{3}$ can be defined geometrically. However, geometry is not as clear in $\mathbb {R}^{n}$ for $n > 3$ . Thus the angle between two nonzero vectors $a, b \in \mathbb {R}^{n}$ is defined to be

\arccos \frac{ \left\langle a, b \right\rangle }{\left\| a\right\| \left\| b\right\| },

where the motivation for this definition comes from the previous exercise. Explain why the Cauchy-Schwarz inequality is needed to show that this definition makes sense.

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Problem 8A.10

(a)(b)

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(a)

Suppose $f$ and $g$ are elements of a real inner product space. Prove that $f$ and $g$ have the same norm if and only if $f+g$ is orthogonal to $f-g$ .

(b)

Use part (a) to show that the diagonals of a parallelogram are perpendicular to each other if and only if the parallelogram is a rhombus.

Problem 8A.11

Suppose $f$ and $g$ are elements of an inner product space. Prove that $\left\| f\right\| = \left\| g\right\|$ if and only if $\left\| sf + tg\right\| = \left\| t f+s g\right\|$ for all $s, t \in \mathbb {R}$ .

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Problem 8A.12

Suppose $f$ and $g$ are elements of an inner product space and $\left\| f\right\| = \left\| g\right\| = 1$ and $\left\langle f, g \right\rangle = 1$ . Prove that $f = g$ .

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Problem 8A.13

Suppose $f$ and $g$ are elements of a real inner product space. Prove that

\left\langle f, g \right\rangle = \frac{\left\| f+g\right\| ^{2}-\left\| f-g\right\| ^{2}}{4}

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Problem 8A.14

Suppose $f$ and $g$ are elements of a complex inner product space. Prove that

\left\langle f, g \right\rangle = \frac{\left\| f+g\right\| ^{2}-\left\| f-g\right\| ^{2}+\left\| f+i g\right\| ^{2} i-\left\| f-i g\right\| ^{2} i}{4}

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Problem 8A.15

Suppose $f, g, h$ are elements of an inner product space. Prove that

\left\| h-\frac{1}{2}(f+g)\right\| ^{2} = \frac{\left\| h-f\right\| ^{2}+\left\| h-g\right\| ^{2}}{2}-\frac{\left\| f-g\right\| ^{2}}{4}

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Problem 8A.16

Prove that a norm satisfying the parallelogram equality comes from an inner product. In other words, show that if $V$ is a normed vector space whose norm $\left\| \cdot \right\|$ satisfies the parallelogram equality, then there is an inner product $\left\langle \cdot , \cdot \right\rangle$ on $V$ such that $\left\| f\right\| = \left\langle f, f \right\rangle^{1 / 2}$ for all $f \in V$ .

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Problem 8A.17

(a)(b)

Let $\lambda$ denote Lebesgue measure on $[1, \infty )$ .

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(a)

Prove that if $f:[1, \infty ) \rightarrow [0, \infty )$ is Borel measurable, then

\left(\int _{1}^{\infty } f(x) d \lambda (x)\right)^{2} \leq \int _{1}^{\infty } x^{2}(f(x))^{2} d \lambda (x)

(b)

Describe the set of Borel measurable functions $f:[1, \infty ) \rightarrow [0, \infty )$ such that the inequality in part (a) is an equality.

Problem 8A.18

(a)(b)

Suppose $\mu$ is a measure. For $f, g \in L^{2}(\mu )$ , define $\left\langle f, g \right\rangle$ by

\left\langle f, g \right\rangle = \int f \overline{g} d \mu

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(a)

Using the inequality

\left|f(x) \overline{g(x)}\right| \leq \frac{1}{2}\left(\left|f(x)\right|^{2}+\left|g(x)\right|^{2}\right)

verify that the integral above makes sense and the map sending $f, g$ to $\left\langle f, g \right\rangle$ defines an inner product on $L^{2}(\mu )$ (without using Hölder's inequality).

(b)

Show that the Cauchy-Schwarz inequality implies that

\left\| f g\right\| _{1} \leq \left\| f\right\| _{2}\left\| g\right\| _{2}

for all $f, g \in L^{2}(\mu )$ (again, without using Hölder's inequality).

Problem 8A.19

Suppose $V_{1}, \ldots , V_{m}$ are inner product spaces. Show that the equation

\left\langle \left(f_{1}, \ldots , f_{m}\right), \left(g_{1}, \ldots , g_{m}\right) \right\rangle = \left\langle f_{1}, g_{1} \right\rangle +\cdots + \left\langle f_{m}, g_{m} \right\rangle

defines an inner product on $V_{1} \times \cdots \times V_{m}$ .

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Remark.

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Each of the inner product spaces $V_{1}, \ldots , V_{m}$ may have a different inner product, even though the same inner product notation is used on all these spaces.

Problem 8A.20

Suppose $V$ is an inner product space. Make $V \times V$ an inner product space as in the exercise above. Prove that the function that takes an ordered pair $(f, g) \in V \times V$ to the inner product $\left\langle f, g \right\rangle \in \mathbf{F}$ is a continuous function from $V \times V$ to $\mathbf{F}$ .

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Problem 8A.21

(a)(b)

Suppose $1 \leq p \leq \infty$ .

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(a)

Show the norm on $\ell^{p}$ comes from an inner product if and only if $p = 2$ .

(b)

Show the norm on $L^{p}(\mathbb {R})$ comes from an inner product if and only if $p = 2$ .

Problem 8A.22

Use inner products to prove Apollonius's identity: In a triangle with sides of length $a, b$ , and $c$ , let $d$ be the length of the line segment from the midpoint of the side of length $c$ to the opposite vertex. Then

a^{2}+b^{2} = \frac{1}{2} c^{2}+2 d^{2}

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