Inner Product Spaces

Prove or give a counterexample: If $v_{1}, \ldots , v_{m} \in V$, then

Suppose $S \in \mathcal{L}(V)$. Define $\left\langle \cdot , \cdot \right\rangle_{1}$ by

for all $u, v \in V$. Show that $\left\langle \cdot , \cdot \right\rangle_{1}$ is an inner product on $V$ if and only if $S$ is injective.

Suppose $T \in \mathcal{L}(V)$ is such that $\left\| T v\right\| \leq \left\| v\right\|$ for every $v \in V$. Prove that $T-\sqrt{2} I$ is injective.

Suppose $V$ is a real inner product space.

Suppose $u, v \in V$. Prove that $\left\langle u, v \right\rangle = 0 \iff \left\| u\right\| \leq \left\| u+a v\right\|$ for all $a \in \mathbf{F}$.

Suppose $u, v \in V$. Prove that $\left\| a u+b v\right\| = \left\| b u+a v\right\|$ for all $a, b \in \mathbb {R}$ if and only if $\left\| u\right\| = \left\| v\right\|$.

Suppose $a, b, c, x, y \in \mathbb {R}$ and $a^{2}+b^{2}+c^{2}+x^{2}+y^{2} \leq 1$. Prove that $a+b+c+4 x+9 y \leq 10$.

Suppose $u, v \in V$ and $\left\| u\right\| = \left\| v\right\| = 1$ and $\left\langle u, v \right\rangle = 1$. Prove that $u = v$.

Suppose $u, v \in V$ and $\left\| u\right\| \leq 1$ and $\left\| v\right\| \leq 1$. Prove that

Find vectors $u, v \in \mathbb {R}^{2}$ such that $u$ is a scalar multiple of $(1,3)$, $v$ is orthogonal to $(1,3)$, and $(1,2) = u+v$.

Suppose $a, b, c, d$ are positive numbers.

Show that the square of an average is less than or equal to the average of the squares. More precisely, show that if $a_{1}, \ldots , a_{n} \in \mathbb {R}$, then the square of the average of $a_{1}, \ldots , a_{n}$ is less than or equal to the average of $a_{1}^{2}, \ldots , a_{n}^{2}$.

Suppose $v \in V$ and $v \neq 0$. Prove that $v /\left\| v\right\|$ is the unique closest element on the unit sphere of $V$ to $v$. More precisely, prove that if $u \in V$ and $\left\| u\right\| = 1$, then

with equality only if $u = v /\left\| v\right\|$.

Suppose $u, v$ are nonzero vectors in $\mathbb {R}^{2}$. Prove that

where $\theta$ is the angle between $u$ and $v$ (thinking of $u$ and $v$ as arrows with initial point at the origin).

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 $x, y \in \mathbb {R}^{n}$ is defined to be

where the motivation for this definition comes from Exercise 15. Explain why the Cauchy-Schwarz inequality is needed to show that this definition makes sense.

Exercise 15 (Exercise 6A.15): if $u, v$ are nonzero vectors in $\mathbb {R}^{2}$, then $\left\langle u, v \right\rangle = \left\| u\right\| \left\| v\right\| \cos \theta$, where $\theta$ is the angle between $u$ and $v$ (thinking of $u$ and $v$ as arrows with initial point at the origin).

Prove that

for all real numbers $a_{1}, \ldots , a_{n}$ and $b_{1}, \ldots , b_{n}$.

Suppose $v_{1}, \ldots , v_{n}$ is a basis of $V$ and $T \in \mathcal{L}(V)$. Prove that if $\lambda$ is an eigenvalue of $T$, then

where $\mathcal{M}(T)_{j, k}$ denotes the entry in row $j$, column $k$ of the matrix of $T$ with respect to the basis $v_{1}, \ldots , v_{n}$.

Prove that if $u, v \in V$, then $\left|\left\| u\right\| -\left\| v\right\| \right| \leq \left\| u-v\right\|$. The inequality above is called the reverse triangle inequality. For the reverse triangle inequality when $V = \mathbb {C}$, see Exercise 2 in Chapter 4.

Exercise 2 in Chapter 4 (Exercise 4.2): if $w, z \in \mathbb {C}$, then $\left|\left|w\right|-\left|z\right|\right| \leq \left|w-z\right|$.

Suppose $u, v \in V$ are such that

What number does $\left\| v\right\|$ equal?

Show that if $u, v \in V$, then

Suppose $v_{1}, \ldots , v_{m} \in V$ are such that $\left\| v_{k}\right\| \leq 1$ for each $k = 1, \ldots , m$. Show that there exist $a_{1}, \ldots , a_{m} \in \left\{ 1,-1\right\}$ such that

Prove or give a counterexample: If $\left\| \cdot \right\|$ is the norm associated with an inner product on $\mathbb {R}^{2}$, then there exists $(x, y) \in \mathbb {R}^{2}$ such that $\left\| (x, y)\right\| \neq \max \left\{ \left|x\right|,\left|y\right|\right\}$.

Suppose $p > 0$. Prove that there is an inner product on $\mathbb {R}^{2}$ such that the associated norm is given by

for all $(x, y) \in \mathbb {R}^{2}$ if and only if $p = 2$.

Suppose $V$ is a real inner product space. Prove that

for all $u, v \in V$.

Suppose $V$ is a complex inner product space. Prove that

for all $u, v \in V$.

A norm on a vector space $U$ is a function

such that $\left\| u\right\| = 0$ if and only if $u = 0,\left\| \alpha u\right\| = \left|\alpha \right|\left\| u\right\|$ for all $\alpha \in \mathbf{F}$ and all $u \in U$, and $\left\| u+v\right\| \leq \left\| u\right\| +\left\| v\right\|$ for all $u, v \in U$. Prove that a norm satisfying the parallelogram equality comes from an inner product (in other words, show that if $\left\| \cdot \right\|$ is a norm on $U$ satisfying the parallelogram equality, then there is an inner product $\left\langle \cdot , \cdot \right\rangle$ on $U$ such that $\left\| u\right\| = \left\langle u, u \right\rangle^{1 / 2}$ for all $u \in U$).

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

defines an inner product on $V_{1} \times \cdots \times V_{m}$. In the expression above on the right, for each $k = 1, \ldots , m$, the inner product $\left\langle u_{k}, v_{k} \right\rangle$ denotes the inner product on $V_{k}$. Each of the spaces $V_{1}, \ldots , V_{m}$ may have a different inner product, even though the same notation is used here.

Suppose $V$ is a real inner product space. For $u, v, w, x \in V$, define

Suppose $u, v, w \in V$. Prove that

Suppose that $E$ is a subset of $V$ with the property that $u, v \in E$ implies $\frac{1}{2}(u+v) \in E$. Let $w \in V$. Show that there is at most one point in $E$ that is closest to $w$. In other words, show that there is at most one $u \in E$ such that

for all $x \in E$.

Suppose $f, g$ are differentiable functions from $\mathbb {R}$ to $\mathbb {R}^{n}$.

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

Fix a positive integer $n$. The Laplacian $\Delta p$ of a twice differentiable realvalued function $p$ on $\mathbb {R}^{n}$ is the function on $\mathbb {R}^{n}$ defined by

The function $p$ is called harmonic if $\Delta p = 0$. A polynomial on $\mathbb {R}^{n}$ is a linear combination (with coefficients in $\mathbb {R}$) of functions of the form $x_{1}^{m_{1}} \ldots x_{n}^{m_{n}}$, where $m_{1}, \ldots , m_{n}$ are nonnegative integers.

Suppose $q$ is a polynomial on $\mathbb {R}^{n}$. Prove that there exists a harmonic polynomial $p$ on $\mathbb {R}^{n}$ such that $p(x) = q(x)$ for every $x \in \mathbb {R}^{n}$ with $\left\| x\right\| = 1$.

Suppose $e_{1}, \ldots , e_{m}$ is a list of vectors in $V$ such that

for all $a_{1}, \ldots , a_{m} \in \mathbf{F}$. Show that $e_{1}, \ldots , e_{m}$ is an orthonormal list. This exercise provides a converse to 6.24.

Suppose $e_{1}, \ldots , e_{m}$ is an orthonormal list in $V$ and $v \in V$. Prove that

Suppose $n$ is a positive integer. Prove that

is an orthonormal list of vectors in $C[-\pi , \pi ]$, the vector space of continuous real-valued functions on $[-\pi , \pi ]$ with inner product

Suppose $f:[-\pi , \pi ] \rightarrow \mathbb {R}$ is continuous. For each nonnegative integer $k$, define

Prove that

The inequality above is actually an equality for all continuous functions $f:[-\pi , \pi ] \rightarrow \mathbb {R}$. However, proving that this inequality is an equality involves Fourier series techniques beyond the scope of this book.

Suppose $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$.

Suppose $T \in \mathcal{L}\left(\mathbb {R}^{3}\right)$ has an upper-triangular matrix with respect to the basis $(1,0,0),(1,1,1),(1,1,2)$. Find an orthonormal basis of $\mathbb {R}^{3}$ with respect to which $T$ has an upper-triangular matrix.

Make $\mathcal{P}_{2}(\mathbb {R})$ into an inner product space by defining $\left\langle p, q \right\rangle = \int_{0}^{1} p q$ for all $p, q \in \mathcal{P}_{2}(\mathbb {R})$.

Suppose $e_{1}, \ldots , e_{m}$ is the result of applying the Gram-Schmidt procedure to a linearly independent list $v_{1}, \ldots , v_{m}$ in $V$. Prove that $\left\langle v_{k}, e_{k} \right\rangle > 0$ for each $k = 1, \ldots , m$.

Suppose $v_{1}, \ldots , v_{m}$ is a linearly independent list in $V$. Explain why the orthonormal list produced by the formulas of the Gram-Schmidt procedure (6.32) is the only orthonormal list $e_{1}, \ldots , e_{m}$ in $V$ such that $\left\langle v_{k}, e_{k} \right\rangle > 0$ and $\operatorname {span}\left(v_{1}, \ldots , v_{k}\right) = \operatorname {span}\left(e_{1}, \ldots , e_{k}\right)$ for each $k = 1, \ldots , m$.

Theorem 6.32 (Gram-Schmidt procedure): suppose $v_{1}, \ldots , v_{m}$ is a linearly independent list of vectors in $V$. Let $f_{1} = v_{1}$. For $k = 2, \ldots , m$, define $f_{k}$ inductively by $f_{k} = v_{k}-\frac{ \left\langle v_{k}, f_{1} \right\rangle }{\left\| f_{1}\right\|^{2}} f_{1}-\cdots -\frac{ \left\langle v_{k}, f_{k-1} \right\rangle }{\left\| f_{k-1}\right\|^{2}} f_{k-1}$. For each $k = 1, \ldots , m$, let $e_{k} = \frac{f_{k}}{\left\| f_{k}\right\| }$. Then $e_{1}, \ldots , e_{m}$ is an orthonormal list of vectors in $V$ such that $\operatorname {span}\left(v_{1}, \ldots , v_{k}\right) = \operatorname {span}\left(e_{1}, \ldots , e_{k}\right)$ for each $k = 1, \ldots , m$.

Find a polynomial $q \in \mathcal{P}_{2}(\mathbb {R})$ such that $p\left(\frac{1}{2}\right) = \int_{0}^{1} p q$ for every $p \in \mathcal{P}_{2}(\mathbb {R})$.

Find a polynomial $q \in \mathcal{P}_{2}(\mathbb {R})$ such that

for every $p \in \mathcal{P}_{2}(\mathbb {R})$.

Show that a list $v_{1}, \ldots , v_{m}$ of vectors in $V$ is linearly dependent if and only if the Gram-Schmidt formula in 6.32 produces $f_{k} = 0$ for some $k \in \left\{ 1, \ldots , m\right\}$.

The Gram-Schmidt formula in 6.32: $f_{1} = v_{1}$ and, for $k = 2, \ldots , m$, inductively $f_{k} = v_{k}-\frac{ \left\langle v_{k}, f_{1} \right\rangle }{\left\| f_{1}\right\|^{2}} f_{1}-\cdots -\frac{ \left\langle v_{k}, f_{k-1} \right\rangle }{\left\| f_{k-1}\right\|^{2}} f_{k-1}$; then $e_{k} = \frac{f_{k}}{\left\| f_{k}\right\| }$.

Suppose $V$ is a real inner product space and $v_{1}, \ldots , v_{m}$ is a linearly independent list of vectors in $V$. Prove that there exist exactly $2^{m}$ orthonormal lists $e_{1}, \ldots , e_{m}$ of vectors in $V$ such that

for all $k \in \left\{ 1, \ldots , m\right\}$.

Suppose $\left\langle \cdot , \cdot \right\rangle_{1}$ and $\left\langle \cdot , \cdot \right\rangle_{2}$ are inner products on $V$ such that $\left\langle u, v \right\rangle_{1} = 0$ if and only if $\left\langle u, v \right\rangle_{2} = 0$. Prove that there is a positive number $c$ such that $\left\langle u, v \right\rangle_{1} = c \left\langle u, v \right\rangle_{2}$ for every $u, v \in V$.

Suppose $V$ is finite-dimensional. Suppose $\left\langle \cdot , \cdot \right\rangle_{1}, \left\langle \cdot , \cdot \right\rangle_{2}$ are inner products on $V$ with corresponding norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$. Prove that there exists a positive number $c$ such that $\left\| v\right\|_{1} \leq c\left\| v\right\|_{2}$ for every $v \in V$.

Suppose $\mathbf{F} = \mathbb {C}$ and $V$ is finite-dimensional. Prove that if $T$ is an operator on $V$ such that 1 is the only eigenvalue of $T$ and $\left\| T v\right\| \leq \left\| v\right\|$ for all $v \in V$, then $T$ is the identity operator.

Suppose $u_{1}, \ldots , u_{m}$ is a linearly independent list in $V$. Show that there exists $v \in V$ such that $\left\langle u_{k}, v \right\rangle = 1$ for all $k \in \left\{ 1, \ldots , m\right\}$.

Suppose $v_{1}, \ldots , v_{n}$ is a basis of $V$. Prove that there exists a basis $u_{1}, \ldots , u_{n}$ of $V$ such that

Suppose $\mathbf{F} = \mathbb {C}, V$ is finite-dimensional, and $\mathcal{E} \subseteq \mathcal{L}(V)$ is such that

for all $S, T \in \mathcal{E}$. Prove that there is an orthonormal basis of $V$ with respect to which every element of $\mathcal{E}$ has an upper-triangular matrix.

Suppose $\mathbf{F} = \mathbb {C}, V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and all eigenvalues of $T$ have absolute value less than 1. Let $\epsilon > 0$. Prove that there exists a positive integer $m$ such that $\left\| T^{m} v\right\| \leq \epsilon \left\| v\right\|$ for every $v \in V$.

Suppose $C[-1,1]$ is the vector space of continuous real-valued functions on the interval $[-1,1]$ with inner product given by

for all $f, g \in C[-1,1]$. Let $\varphi$ be the linear functional on $C[-1,1]$ defined by $\varphi (f) = f(0)$. Show that there does not exist $g \in C[-1,1]$ such that

for every $f \in C[-1,1]$. This exercise shows that the Riesz representation theorem (6.42) does not hold on infinite-dimensional vector spaces without additional hypotheses on $V$ and $\varphi$.

For all $u, v \in V$, define $d(u, v) = \left\| u-v\right\|$.

Suppose $v_{1}, \ldots , v_{m} \in V$. Prove that

Suppose $U$ is a subspace of $V$ with basis $u_{1}, \ldots , u_{m}$ and

is a basis of $V$. Prove that if the Gram-Schmidt procedure is applied to the basis of $V$ above, producing a list $e_{1}, \ldots , e_{m}, f_{1}, \ldots , f_{n}$, then $e_{1}, \ldots , e_{m}$ is an orthonormal basis of $U$ and $f_{1}, \ldots , f_{n}$ is an orthonormal basis of $U^{\perp }$.

Suppose $U$ is the subspace of $\mathbb {R}^{4}$ defined by

Find an orthonormal basis of $U$ and an orthonormal basis of $U^{\perp }$.

Suppose $e_{1}, \ldots , e_{n}$ is a list of vectors in $V$ with $\left\| e_{k}\right\| = 1$ for each $k = 1, \ldots , n$ and

for all $v \in V$. Prove that $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$. This exercise provides a converse to $6.30(b)$.

Suppose that $V$ is finite-dimensional and $U$ is a subspace of $V$. Show that $P_{U^{\perp }} = I-P_{U}$, where $I$ is the identity operator on $V$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Show that