Continuous Linear Operators between Banach Spaces

Prove Proposition 11.1.10.

Proposition 11.1.10: Let $X, Y$ be normed linear spaces. The operator norm on $\mathcal{L}(X, Y)$ is a norm.

Prove Proposition 11.1.11.

Proposition 11.1.11: Let $X, Y$ be normed linear spaces and let $T \in \mathcal{L}(X, Y)$. Then (a) $\left\| T\right\| = \sup \left\{ \left\| Tx\right\| : \left\| x\right\| \leq 1\right\} = \sup \left\{ \left\| Tx\right\| : \left\| x\right\| = 1\right\}$. (b) $\left\| Tx\right\| \leq \left\| T\right\| \left\| x\right\|$, for all $x \in X$.

Let $X, Y$ be normed linear spaces and $T \in \mathcal{L}(X, Y)$. Suppose that $M>0$ and $\left\| T\right\| \leq M$.

Let $X, Y$ be normed linear spaces, and let $T: X \rightarrow Y$ be a linear operator. Prove that $T$ is bounded if and only if it maps bounded sets to bounded sets.

Let $X$ be a normed linear space and $x_{0} \in X$. Prove that a linear operator on a normed space is continuous if and only if it is continuous at $x_{0}$.

Let $X$ be a normed linear space and let $T, S \in \mathcal{L}(X, X)$. Prove that the composition $S T$ belongs to $\mathcal{L}(X, X)$ and that $\left\| S T\right\| \leq \left\| S\right\| \left\| T\right\|$. (We write $S T$ for $S \circ T$ just like when $S$ and $T$ are matrices.)

Let $X$ be a Banach space and $\mathcal{F}$ a family of bounded linear operators from $X$ to a normed linear space $Y$. Prove that $\mathcal{F}$ is equicontinuous if and only if it is uniformly bounded.

Let $\mathcal{H}$ be Hilbert space and $T \in \mathcal{L}(\mathcal{H})$. Prove that

Let $X, Y$ be normed linear spaces and $T: X \rightarrow Y$ a linear map such that $\left\{ T x_{n}\right\}$ is bounded in $Y$ whenever the sequence $\left\{ x_{n}\right\}$ converges to 0 in $X$. Prove that $T \in \mathcal{L}(X, Y)$.

Let $X, Y$ be normed linear spaces, and let $T: X \rightarrow Y$ be a linear operator. Prove that $T$ is bounded if and only if it maps every Cauchy sequence in $X$ to a Cauchy sequence in $Y$.

Prove that the set of all linear transformations from a vector space $X$ to a vector space $Y$ is a vector space.

Let $Z$ be a subspace of a normed linear space $X$, let $Y$ be a Banach space, and let $A: Z \rightarrow Y$ be a continuous linear operator. Prove that there exists a unique continuous linear operator $\bar{A}: \bar{Z} \rightarrow Y$ such that $\bar{A} \mid Z=A$ and $\left\| \bar{A}\right\| =\left\| A\right\|$.

Let $X, Y$ be normed spaces, let $T \in \mathcal{L}(X, Y)$, and let $\tilde{T}: X / \operatorname {ker} T \rightarrow Y$ be defined by $\tilde{T}([x])=T x$ (see Exercise 10.1.13). Prove that $\tilde{T}$ is a welldefined, injective, bounded, linear operator for which $\left\| \tilde{T}\right\| =\left\| T\right\|$.

Let $X$ be a normed space and $V$ a closed subspace of $X$ (see Exercise 10.1.13). Prove that:

Let $\mathcal{H}$ be Hilbert space and let $T \in \mathcal{L}(\mathcal{H})$. Prove that $T=0$ if and only if $\left\langle T x, y \right\rangle =0$ for all $x, y \in \mathcal{H}$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X, X)$ with $\left\| T\right\| <1$. Define $T^{0}=\operatorname {Id}$. Prove that $\sum_{n=0}^{\infty } T^{n}$ converges in $\mathcal{L}(X, X)$.

Let $X, Y$ be normed linear spaces, let $\left\{ T_{n}\right\}$ converge to $T$ in $\mathcal{L}(X, Y)$, and let $\left\{ x_{n}\right\}$ converge to $x$ in $X$. Prove that $\left\{ T_{n} x_{n}\right\}$ converges to $T x$ in $Y$.

Let $X, Y$ be normed linear spaces and $\left\{ T_{n}\right\} , T \in \mathcal{L}(X, Y)$. Prove that $\lim_{n} T_{n}=T$ if and only if $\left\{ T_{n} x\right\}$ converges to $T x$ uniformly on the unit ball of $X$.

Let $g$ be a continuous function on $[0,1]$ and let $T: L^{p}\left({[0,1]}\right) \rightarrow L^{p}\left({[0,1]}\right)$ be defined by $T(f)=g f$. Prove that $T$ is a bounded operator and $\left\| T\right\| =\left\| g\right\|_{\max }$.

Let $p, q$ be real numbers such that $q>1$ and $1 / p+1 / q=1$, and let $\left(a_{i j}\right)$ be an infinite matrix of real numbers such that

Prove that

defines a bounded linear operator

from $\ell^{p}$ to $\ell^{q}$.

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

Prove that

defines an operator in $\mathcal{L}\left(L^{\infty }\right)$,

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a continuous function. Prove that

defines an operator in $\mathcal{L}(C[0,1])$,

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

where $\left\| K(s, t)\right\|_{t, \infty }$ means the $L^{\infty }$ norm of $K$ as a function of $t$ for a fixed $s$. Prove that

defines an operator in $\mathcal{L}\left(L^{1}\left({[0,1]}\right)\right)$,

Let $q>1$ and let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

Prove that

defines an operator in $\mathcal{L}\left(L^{p}\left({[0,1]}\right), L^{q}\left({[0,1]}\right)\right),(1 / p+1 / q=1)$

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

Prove that

defines an operator in $\mathcal{L}\left(L^{2}\left({[0,1]}\right)\right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $h$ be a bounded measurable function on $X$, and let $T$ be the operator on $L^{p}\left(X, \mu \right)$ defined by $T f=h f$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $p \geq q$. Find the norm of the identity operator $\operatorname {Id}: L^{p}\left(X, \mu \right) \rightarrow L^{q}\left(X, \mu \right)$.

Let $1 \leq p \leq q<\infty$. Determine for what functions $h$ on $[0,1]$ is the multiplication by $h$ a bounded operator from $L^{p}[0,1]$ to $L^{q}[0,1]$.

Let $T: C[0,1] \rightarrow C[0,1]$ be defined by $T f(x)=x^{3} f(x)$, for all $f \in C[0,1]$ and each $x \in [0,1]$.

Let $T: C[0,1] \rightarrow C[0,1]$ be defined by $T f(x)=\int_{0}^{x} f(t) d t$, for all $f \in C[0,1]$ and each $x \in [0,1]$. Prove that $T$ is a bounded linear operator and find its norm.

Let $T: L^{1}[0,1] \rightarrow L^{1}[0,1]$ be defined by $T f(x)=f(\sqrt{x})$, for all $f \in C[0,1]$ and each $x \in [0,1]$. Prove that $T$ is a bounded linear operator and find its norm.

For $f \in L^{1}[0,1]$ let $x_{n}=\int_{0}^{1} f(t) t^{n} d t$, and let $T(f)=\left\{ x_{n}\right\}$. Prove that the operator $T$ is a bounded linear operator mapping $L^{1}[0,1]$ to $c_{0}$ and determine its norm. (See Exercise 10.1.15.)

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \nu$ ) be $\sigma$-finite measure spaces, let $K$ be an $\mathfrak {M} \otimes \mathfrak {N}$ measurable function on $X \times Y$ and let $C>0$. Suppose that

Prove that the operator $T$ defined on $L^{p}\left(X, \mu \right)$ by

is a bounded operator and that $\left\| T f\right\|_{p} \leq C\left\| f\right\|_{p}$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $1 \leq p<r \leq \infty$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $1 \leq p<r \leq \infty$. Prove that:

Let $X, Y$ be Banach spaces and let $S, T \in \mathcal{L}(X, Y)$. Suppose that the operator $S T$ is invertible. Prove or disprove: $T$ is invertible.

Let $X$ be a Banach space and let $T \in \mathcal{L}(X)$ satisfy $\left\| T\right\| \leq \eta <1$. Prove that the operator $\operatorname {Id}-T$ has a bounded inverse $(\operatorname {Id}-T)^{-1}$ and that $\left\| (\operatorname {Id}-T)^{-1}\right\| \leq 1 /(1-\eta )$.

Let $X, Y$ be normed spaces and $T: X \rightarrow Y$ a linear map that has a unique left inverse $S$. Prove that $T$ is invertible.

Give an example of normed spaces $X, Y$ and an operator $T: X \rightarrow Y$ that is:

Let $X$ be a linear space and let $S, T: X \rightarrow X$ be linear maps. Prove:

Let $X$ be a linear space and let $T: X \rightarrow X$ be a linear map that satisfies equation $p(T)=0$ for some polynomial $p$ with $p(0) \neq 0$. Prove that $T$ is invertible.

Let $X$ be a linear space, let $T: X \rightarrow X$ be a bounded map that satisfies equation $T^{2}=T$, and let $0<\left|\lambda \right|<1$. Find the inverse of $\lambda \operatorname {Id}-T$.

Let $X$ be a linear space, let $T: X \rightarrow X$ be a bounded operator with $\left\| T\right\| <1$. Prove or disprove: the equation $x-T x=y$ has a solution $x$ for every $y \in X$.

Let $X$ be a Banach space. Prove that the set of invertible operators in $\mathcal{L}(X, X)$ is open.

Let $\mathcal{H}$ be Hilbert space, let $\left\{ x_{n}\right\}_{n=1}^{\infty }$ be an orthonormal basis of $\mathcal{H}$, let $0 \leq C<1$, and let $\left\{ y_{n}\right\}_{n=1}^{\infty }$ be a sequence in $\mathcal{H}$ such that for any scalars $\left\{ \alpha_{n}\right\}$,

Prove that for each $x \in \mathcal{H}$ there are scalars $\left\{ \beta_{n}\right\}$ such that $x=\sum_{k=1}^{\infty } \beta_{k} y_{k}$, with the series converging in the norm of $\mathcal{H}$.

Let $X$ be a normed linear space of dimension $n$, let $\left\{ e_{1}, \ldots , e_{n}\right\}$ be a basis of $X$, and let $T: \mathbb {R}^{n} \rightarrow (X,\left\| \cdot \right\| )$ be defined by $T\left(x_{1}, \ldots , x_{n}\right)= \sum_{k=1}^{n} x_{k} e_{k}$. Prove that $T$ is an invertible linear operator.

Prove Corollary 11.2.6.

Corollary 11.2.6: Two normed linear spaces are isomorphic if and only if there is a linear operator $T$ mapping $X$ onto $Y$ and positive constants $m, M$ such that $m\left\| x\right\| \leq \left\| Tx\right\| \leq M\left\| x\right\|$, for all $x \in X$.

Prove Corollary 11.2.7.

Corollary 11.2.7: Let $X$ be a vector space and let $\left\| \cdot \right\|_{\alpha }$ and $\left\| \cdot \right\|_{\beta }$ be two norms on $X$. Normed spaces $(X, \left\| \cdot \right\|_{\alpha })$ and $(X, \left\| \cdot \right\|_{\beta })$ are isomorphic if and only if there are positive constants $m, M$ such that $m\left\| x\right\|_{\alpha } \leq \left\| x\right\|_{\beta } \leq M\left\| x\right\|_{\alpha }$, for all $x \in X$.

Prove that any two infinite-dimensional separable Hilbert spaces are isometrically isomorphic and that any such isomorphism preserves the inner product.

Prove that the space $C[0,1]$ is isometrically isomorphic to its subspace $A=\left\{ f \in C[0,1]: f(t)=f(1-t)\right\}$.

Let $B\left(\ell^{1}\right)$ denote the vector space of bounded sequences $\left\{ a_{n}\right\}$, where for each $n \in \mathbb {N}, a_{n} \in \ell^{1}$. The norm on $B\left(\ell^{1}\right)$ is defined by

Prove that the normed spaces $B\left(\ell^{1}\right)$ and $\mathcal{L}\left(\ell^{1}\right)$ are isometrically isomorphic.

Let $X, Y$ be Banach spaces and let $T \in \mathcal{L}(X, Y)$ be an isometry. Prove that the range of $T$ is a closed subspace of Y .

Let $X, Y, Z$ be normed linear spaces, let $F: X \rightarrow Y$ be an isometric isomorphism, and let $G \in \mathcal{L}(Y, Z)$. Prove that $\left\| G\right\| =\left\| G \circ F\right\|$.

Let $\mathcal{H}$ be Hilbert space. Prove that there is a measure space ( $X, \mathfrak {M}, \mu$ ) for which $\mathcal{H}$ is isometrically isomorphic to $L^{2}\left(X, \mu \right)$.

Let $\mathcal{H}$ be an infinite dimensional Hilbert space. Prove that there exists a continuous injective mapping $\gamma :[0,1] \rightarrow \mathcal{H}$ such that $\gamma (b)-\gamma (a)$ is orthogonal to $\gamma (d)-\gamma (c)$ whenever $0 \leq a \leq b \leq c \leq d \leq 1$.

Prove that if $T \in \mathcal{L}(\mathcal{H})$ then $T$ is an isometry on Hilbert space $\mathcal{H}$ if and only if $\left\langle T u, T v \right\rangle = \left\langle u, v \right\rangle$ for all $u, v \in \mathcal{H}$.

Let $X$ be a vector space and let $\left\| \cdot \right\|_{\alpha }$ and $\left\| \cdot \right\|_{\beta }$ be two equivalent norms on $X$. Prove that a sequence $\left\{ x_{n}\right\}_{n=1}^{\infty }$ in $X$ is Cauchy with respect to $\left\| \cdot \right\|_{\alpha }$ if and only if it is Cauchy with respect to $\left\| \cdot \right\|_{\beta }$. Conclude that ( $X,\left\| \cdot \right\|_{\alpha }$ ) is complete if and only if ( $X,\left\| \cdot \right\|_{\beta }$ ) is.

Let $X$ be a vector space and let $\left\| \cdot \right\|_{\alpha }$ and $\left\| \cdot \right\|_{\beta }$ be two equivalent norms on $X$. Let $\mathcal{T}_{\alpha }$ and $\mathcal{T}_{\beta }$ be the corresponding topologies. Prove that $\mathcal{T}_{\alpha }=\mathcal{T}_{\beta }$.

Prove that the equivalence of norms is an equivalence relation.

Let $X$ be a normed linear space and let $T$ be a linear invertible the operator on $X$ such that

Prove that there exists an equivalent norm on $X$ which makes $T$ an isometry.

Prove that $\left\| f\right\| =\left|f(0)\right|+\left\| f^{\prime }\right\|_{\infty }$ defines a norm on $C^{1}[0,1]$ that is equivalent to the norm defined by $\left\| f\right\|_{\alpha }:=\left\| f\right\|_{\infty }+\left\| f^{\prime }\right\|_{\infty }$.

Let $X$ be a linear space equipped with two norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ such that ( $X,\left\| \cdot \right\|_{1}$ ) is complete and the identity mapping $\operatorname {Id}:\left(X,\left\| \cdot \right\|_{1}\right) \rightarrow \left(X,\left\| \cdot \right\|_{2}\right)$ is continuous. Suppose that there exists $r>0$ such that $B_{2}(0,1) \subset B_{1}(0, r)+ B_{2}(0,1 / 2)$. Prove that these norms are equivalent.

Prove that there is $M>0$ such that if $p(t)=a_{0}+a_{1} t+\cdots +a_{n} t^{n}$ then

Let

Given $p \in [1, \infty )$, define

Show that $\left\| \cdot \right\|_{p}^{i n f}$ is a norm for each $p$, and all of these norms are equivalent.

Give an example of an open cover of the closed unit ball in $\ell^{2}$ that has no finite subcover.

Give an example of an open cover of the closed unit ball in $C[0,1]$ that has no finite subcover.

The Hilbert cube is a subset of $\ell^{2}$ consisting of all sequences $\left\{ x_{n}\right\}$ such that $\left|x_{n}\right| \leq 1 / n$, for all $n \in \mathbb {N}$. Prove that it is a compact set.

Let $\left\{ e_{n}\right\}$ be an orthonormal basis of Hilbert space $\mathcal{H}$, let $\left\{ a_{n}\right\} \subset$ ( $0,+\infty$ ), and let

Prove that $A$ is compact if and only if $\sum_{n} a_{n}^{2}$ is convergent.

Let $X$ be an infinite dimensional normed linear space. Prove that there exists a sequence $\left\{ x_{n}\right\} \subset X$ such that $\left\| x_{n}\right\| =1$ for all $n \in \mathbb {N}$ and $\left\| x_{m}-x_{n}\right\| \geq 1 / 2$ for $m \neq n$.

Prove that the vector space of all polynomials in one variable is not a Banach space in any norm.

Let $X, Y$ be normed linear spaces. Prove that:

Let $\mathcal{F}$ be a closed subspace of $L^{p}\left(\mathbb {R}, \mathcal{M}, m\right), 1 \leq p<\infty$. Prove that $\mathcal{F}$ is compact if and only if all of the following conditions hold:

Let $X$ be a Banach space and $Y$ a normed linear space, and let $T_{n}$ : $X \rightarrow Y, n \in \mathbb {N}$, be bounded linear operators. Suppose that, for every $x \in X$, the sequence $T_{n} x$ is convergent, and let $T$ be an operator defined by $T x= \lim T_{n} x$. Prove that $T$ is a bounded linear operator.

Let $X$ be a Banach space and $\mathcal{F}$ an unbounded family of bounded linear operators from $X$ to a normed linear space $Y$. Prove that the set of $x \in X$ for which $\sup \left\{ \left\| T x\right\| : T \in \mathcal{F}\right\} =\infty$ is dense in $X$.

Let $X, Y, Z$ be Banach spaces and let $F: X \times Y \rightarrow Z$ be a separately continuous bilinear map, i.e., $F_{x} \in \mathcal{L}(Y, Z)$ for each $x \in X$ and $F^{y} \in \mathcal{L}(X, Z)$ for each $y \in Y$. Prove that $F$ is continuous.

Give an example to show that the Uniform Boundedness Principle (Theorem 11.3.1) does not hold if $X$ is not a Banach space.

Theorem 11.3.1 (The Uniform Boundedness Principle): Let $X$ be a Banach space and $\mathcal{F}$ a family of bounded linear operators from $X$ to a normed linear space $Y$. If, for every $x \in X$, $\sup \left\{ \left\| Tx\right\| : T \in \mathcal{F}\right\} < \infty$, then $\sup \left\{ \left\| T\right\| : T \in \mathcal{F}\right\} < \infty$.

Let $X, Y$ be Banach spaces and $\left\{ T_{j, k}\right\} \subset \mathcal{L}(X, Y)$. Suppose that for each $k \in \mathbb {N}$ there is $x \in X$ such that $\sup_{j}\left\| T_{j, k} x\right\| =\infty$. Prove that there exists $x \in X$ such that $\sup_{j}\left\| T_{j, k} x\right\| =\infty$ for all $k \in \mathbb {N}$.

If $X$ is a Banach space the sequence $\left\{ T_{n}\right\} \subset \mathcal{L}(X)$ converges strongly to a linear operator $T: X \rightarrow X$ if $\lim T_{n} x=T x$ for all $x \in X$. If $\mathcal{H}$ is Hilbert space the sequence $\left\{ T_{n}\right\} \subset \mathcal{L}(\mathcal{H})$ converges weakly to a linear operator $T: \mathcal{H} \rightarrow \mathcal{H}$ if $\lim \left\langle T_{n} x, y \right\rangle = \left\langle T x, y \right\rangle$ for all $x, y \in \mathcal{H}$.