Math Problems Vault

Problem 44

(a)(b)

Let $\left(X,\left\| .\right\|_{1}\right)$ and $\left(Y,\left\| .\right\|_{2}\right)$ be norm spaces, and $T \in \mathcal{B}(X, Y)$ . We define $\left\| T\right\|$ as

\left\| T\right\| : = \inf _{x \in X}\left\{ M:\left\| T x\right\| _{2} \leq M\left\| x\right\| _{1}\right\} .

?

(a)

Show that

\left\| T\right\| = \sup _{\left\| x\right\| _{1} = 1}\left\| T x\right\| _{2} = \sup _{\left\| x\right\| _{1} \leq 1}\left\| T x\right\| _{2} = \sup _{x \in X, x \neq 0} \frac{\left\| T x\right\| _{2}}{\left\| x\right\| _{1}}.

(b)

Show that

\left\| T\right\| = \sup _{\left\| x\right\| _{1} < 1}\left\| T x\right\| _{2}.

Problem 46

(a)(b)

Let $H = \ell^{2}$ be the well known Hilbert space. Consider the left shift defined by

L: \ell ^{2} \rightarrow \ell ^{2}, \quad x = \left(x_{1}, x_{2}, x_{3}, \ldots \right) \mapsto L x = \left(x_{2}, x_{3}, \ldots \right).

?

(a)

Show that $L$ is a linear bounded operator. Find $\left\| L\right\|$ .

(b)

Define the right shift and answer the same question as in part (a).

Problem 47

Let $X = C[0,1]$ with the max-norm (the uniform norm). We define the integral operator

K: X \rightarrow X \quad \text{ by } \quad K f(x) = \int _{0}^{x} f(y) d y.

Show that $K$ is bounded. Find $\left\| K\right\|$ .

?

Problem 48

Let $X = L^{2}[0,1]$ with the norm $\left\| .\right\|_{2}$ . We define the integral operator

A: X \rightarrow X \quad \text{ by } \quad A f(x) = \int _{0}^{x} f(y) d y

Show that $A$ is bounded. Find $\left\| A\right\|$ .

?

Problem 49

(a)(b)

Let $a, b$ be real numbers such that $a < b$ . Consider the Hilbert space $L^{2}[a, b]$ over $\mathbb {R}$ and the operator $T: L^{2}[a, b] \rightarrow \mathbb {R}$ defined by

T f = \int _{a}^{b} f(x) d x, \quad f \in L^{2}[a, b]

?

(a)

Show that $T$ is bounded. Compute $\left\| T\right\|$ .

(b)

According to the Riesz's Theorem, there exists a function $g \in L^{2}[a, b]$ such that

T f = \langle f, g\rangle \text{ for all } f \in L^{2}[a, b]

Find such a function $g$ and verify that $\left\| g\right\|_{L^{2}} = \left\| T\right\|$ .

Problem 50

Let $X, Y$ be normed spaces and $T \in \mathcal{B}(X, Y)$ . Consider the following statement:

T \text{ is an isometry } \Leftrightarrow \left\| T\right\| = 1 \text{. }

Do you agree with it? Why?

?

Problem 51

(a)(b)

Define $T: C[0,1] \rightarrow C[0,1]$ by

(T x)(t) = t \int _{0}^{t} x(s) d s

?

(a)

Prove that $T$ is a bounded linear operator. Compute $\left\| T\right\|$ .

(b)

Prove that the inverse $T^{-1}: \operatorname {Image}(T) \rightarrow C[0,1]$ exists but not bounded.

Problem 53

(a)(b)(c)

Let $1 < p < \infty$ and $q$ be its exponent conjugate. For $f \in L^{p}(0, \infty )$ , let

(T f)(x) = \frac{1}{x} \int _{0}^{x} f(t) d t = \int _{0}^{1} f(t x) d t

Show that

?

(a)

$T f$ is well-defined on $(0, \infty )$ , and that $T f$ is continuous with respect to $x$ .

(b)

$T f \in L^{p}(0, \infty )$ .

(c)

$T$ is a bounded linear operator from $L^{p}(0, \infty )$ to itself. Calculate its norm. (Hint: Use $f_{n} = x^{-1 / p} \chi_{\left\{ 1 \leq x \leq n\right\} }$ .)

Problem 54

Let $\left(c_{j}\right)_{j = 1}^{\infty }$ be a sequence of complex numbers. Define an operator $D$ on $\ell^{2}$ by

D x = \left(c_{1} x_{1}, c_{2} x_{2}, \ldots \right) \text{ for } x = \left(x_{1}, x_{2}, \ldots \right) \in \ell ^{2}.

Prove that $D$ is bounded if and only if $\left(c_{j}\right)_{j = 1}^{\infty }$ is bounded, and in this case $\left\| D\right\| = \sup_{j}\left|c_{j}\right|$ .

?

Problem 55

Prove that $\mathcal{B}(\mathbb {F}, Y)$ is not a Banach space if $Y$ is not complete. Hint: Take a Cauchy sequence $\left(y_{n}\right)$ in $Y$ which does not converge and consider the sequence of operators $(B_{n})$ defined by :

B_{n} \lambda : = \lambda y_{n} ; \quad \lambda \in \mathbb {F}.

?

Problem 56

Let $T_{1}, T_{2}, \ldots$ be the following bounded linear operators $\ell^{1} \rightarrow \ell^{\infty }$ :

\begin{aligned} & T_{1}\left(x_{1}, x_{2}, x_{3}, \ldots \right) = \left(x_{1}, x_{1}, x_{1}, x_{1}, \ldots \right) \\ & T_{2}\left(x_{1}, x_{2}, x_{3}, \ldots \right) = \left(x_{1}, x_{2}, x_{2}, x_{2}, \ldots \right) \\ & T_{3}\left(x_{1}, x_{2}, x_{3}, \ldots \right) = \left(x_{1}, x_{2}, x_{3}, x_{3}, \ldots \right) \ldots \text{ etc } \end{aligned}

Prove that the sequence $(T_{n})$ is strongly operator convergent. Also prove that $\left(T_{n}\right)$ is not uniformly operator convergent.

?

Problem 57

Let $H_{1}$ and $H_{2}$ be two Hilbert spaces. Let $\left\{ a_{1}, \ldots , a_{n}\right\}$ be an orthonormal system of $H_{1}$ and $\left\{ b_{1}, \ldots , n\right\}$ be an orthonormal system of $H_{2}$ , and $\lambda_{1}, \ldots , \lambda_{n} \in \mathbb {K}$ . Consider the operator

U: H_{1} \rightarrow H_{2} \quad \text{ defined by } \quad U(x) = \sum _{i = 1} \lambda _{i} b_{i} \left\langle x, a_{i} \right\rangle .

Calculate $\left\| U\right\|$ .

?

Problem 58

(a)(b)

?

(a)

Let $X$ be a Hilbert space. Let $a, b \in H$ be two non-zero orthogonal elements. Consider the operator

U: H \rightarrow H, \quad U(x) = a\langle x, b\rangle +b\langle x, a\rangle .

Calculate $\left\| U\right\|$ .

(b)

Consider the operator $T: L^{2}[0, \pi ] \rightarrow L^{2}[0, \pi ]$ defined by

(T f)(x) = \sin x \int _{0}^{\pi } f(t) \cos t d t+\cos x \int _{0}^{\pi } f(t) \sin t d t

Calculate $\left\| T\right\|$ .

Problem 59

(a)(b)

?

(a)

Let $H$ be a Hilbert space and $\left\{ e_{1}, e_{2}\right\} \subset H$ an orthonormal system. Let

A = \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]

be scalar square matrix. Consider the operators $U, V: H \rightarrow H$ defined by

\begin{aligned} & U(x) = a \left\langle x, e_{1} \right\rangle e_{1}+b \left\langle x, e_{2} \right\rangle e_{2} \\ & V(x) = c \left\langle x, e_{1} \right\rangle e_{1}+d \left\langle x, e_{2} \right\rangle e_{2} \end{aligned}

Prove that

\left\| U+V\right\| ^{2}+\left\| U-V\right\| ^{2} = 2\left(\left\| U\right\| ^{2}+\left\| V\right\| ^{2}\right)

if and only if

(\max \left\{ \left|a+c\right|,\left|b+d\right|\right\} )^{2}+(\max \left\{ \left|a-c\right|,\left|b-d\right|\right\} )^{2} = 2\left(\max \left\{ \left|a\right|,\left|b\right|\right\} ^{2}+\max \left\{ \left|c\right|,\left|d\right|\right\} ^{2}\right).

(b)

Prove that if $\operatorname {dim} H \geq 2$ then $\mathcal{B}(H)$ is not a Hilbert space

Problem 60

The dual space of $\ell^{1}$ is $\ell^{\infty }$ , that is, $\left(\ell^{1}\right)^{*} = \ell^{\infty }$ .

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Problem 61

The dual space of $\ell^{p}$ is $\ell^{q}$ , that is, $\left(\ell^{p}\right)^{*} = \ell^{q}$ , here $1 < p < \infty$ and $\frac{1}{p}+\frac{1}{q} = 1$ .

?

Problem 62

The dual space of $c_{0}$ is $\ell^{1}$ , that is, $\left(c_{0}\right)^{*} = \ell^{1}$ .

?

Problem 63

Let $X$ be a normed space and $f, g$ are nonzero linear functionals on $X$ . Show that

\operatorname {ker}(f) = \operatorname {ker}(g) \Longleftrightarrow f = c g \quad \text{ for some nonzero scalar } c.

?

Problem 64

Let $X$ be a normed space and $f$ are nonzero linear functional on $X$ . Show that $f$ is continuous if and only if $\operatorname {ker}(f)$ is closed.

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Problem 65

Let $Z$ be a subspace of a normed space $X$ , and $y \in X$ . Let $d = d(y, Z)$ . Prove that there exists $\Lambda \in X^{*}$ such that $\left\| \Lambda \right\| \leq 1, \Lambda (y) = d$ and $\Lambda (z) = 0$ for all $z \in Z$ .

?

Problem 66

Let $X$ be a normed space and $\left(x_{n}\right)$ be a sequence in $X$ . Set $V: = \operatorname {Span}\left\{ x_{1}, x_{2}, \ldots \right\}$ . Let $W$ be the set of all continuous $f \in X^{*}$ such that $f\left(x_{n}\right) = 0, \forall n \in \mathbb {N}$ . Prove that

\bar{V} = \left\{ x \in X: f(x) = 0, \forall f \in W\right\} .

?

Problem 67

Let $X$ be a normed space and $\left\{ x_{1}, \ldots , x_{n}\right\} \subset X$ a linearly independent system. Prove that for any $\alpha_{1}, \ldots , \alpha_{n} \in \mathbb {K}$ there exists $x^{*} \in X^{*}$ such that

x^{*}\left(x_{i}\right) = \alpha _{i}, \forall i \in \left\{ 1, \ldots , n\right\}

?

Problem 68

(a)(b)(c)

Let $X$ be a normed space and $Y \subset X$ a linear subspace. For $x_{0} \in X \backslash Y$ we define

f: \operatorname {Span}\left\{ Y, x_{0}\right\} \rightarrow \mathbb {K}, f\left(y+\lambda x_{0}\right) = \lambda , \forall y \in Y, \forall \lambda \in \mathbb {K}.

?

(a)

Prove that $f$ is well defined and linear.

(b)

Prove that $f$ is continuous if and only if $x_{0} \notin \bar{Y}$ .

(c)

Prove that $\bar{Y} = \bigcap \left\{ \operatorname {ker} x^{*}: x^{*} \in X^{*}, Y \subset \operatorname {ker} x^{*}\right\}$ .