The Hahn-Banach Theorem

Let $\mathcal{M}=\left\{ (x, y) \in \mathbb {R}^{2}: 2 x-y=0\right\}$ and let $T: \mathcal{M} \rightarrow \mathbb {R}$ be defined by $f(x, y)=x$. Show that $T^{\prime }: \mathbb {R}^{2} \rightarrow \mathbb {R}$ defined by $T^{\prime }(x, y)=x / 5+2 y / 5$ is the unique Hahn-Banach extension of $T$.

Let $\mathcal{M}=\left\{ \left\{ x_{n}\right\} : x_{1}=x_{3}=x_{5}=\cdots =0\right\} \subset \ell^{1}$. Show that every non-zero functional $T: \mathcal{M} \rightarrow \mathbb {R}$ has infinitely many Hahn-Banach extensions to $\ell^{1}$.

Let $Y$ be the subspace of $C[a, b]$ consisting of all constant function and let $T \in Y^{*}$ be defined by $T f=f(a)$, for all $f \in Y$. Define

Prove that $T^{\prime }$ is a Hahn-Banach extension of $T$.

Prove that the function $T_{0}$ defined in the proof of the Hahn-Banach theorem is a well-defined, linear functional on the set $Z$ satisfying $\left\| T_{0}\right\|_{Z^{*}} \leq \left\| T\right\|_{Y^{*}}$.

The setting, from the proof of Theorem 12.2.2 (The Hahn-Banach Theorem: if $X$ is a normed linear space, $Y \subset X$ a linear subspace and $T$ a bounded linear functional on $Y$, then there is a linear functional $\tilde{T}$ on $X$ with $\tilde{T} x=T x$ for $x \in Y$ and $\left\| \tilde{T}\right\| =\left\| T\right\|$): let $\mathcal{A}$ be the collection of all pairs $(T^{\prime }, Y^{\prime })$, where $Y^{\prime } \supset Y$ is a linear subspace of $X$ and $T^{\prime }$ is an extension of $T$ to $Y^{\prime }$ that does not increase the norm, partially ordered by $(T^{\prime }, Y^{\prime }) \prec (T^{\prime \prime }, Y^{\prime \prime })$ if $Y^{\prime } \subset Y^{\prime \prime }$ and $T^{\prime \prime }=T^{\prime }$ on $Y^{\prime }$. Let $\mathcal{A}_{0}$ be a totally ordered subfamily of $\mathcal{A}$ and $Z=\cup \left\{ Y^{\prime }: (T^{\prime }, Y^{\prime }) \in \mathcal{A}_{0}\right\}$. For $z \in Z$ choose $(T^{\prime }, Y^{\prime }) \in \mathcal{A}_{0}$ with $z \in Y^{\prime }$ and define $T_{0}(z)=T^{\prime }(z)$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=a_{0}$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=\sum_{k=0}^{\operatorname {deg} p} a_{k}$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=\sum_{k=0}^{\operatorname {deg} p}(-1)^{k} a_{k}$.

In Exercises 12.2.5-12.2.8 $\mathcal{P}$ is the vector space of polynomials defined on $[0,1]$ and $p(x)=\sum_{k=0}^{\operatorname {deg} p} a_{k} x^{k}$. Determine which of the following linear functionals on $\mathcal{P}$ can be extended to a bounded linear functional on $C[0,1]$: $T p=\sum_{k=0}^{N} c_{k} a_{k}$, where $N$ is a fixed positive integer and $\left\{ c_{k}\right\}_{k=1}^{N}$ are fixed real numbers.

Let $X$ be a normed linear space and let $Y \subset X$ be a finite-dimensional linear subspace. Prove that there exists a closed subspace $Z \subset X$ such that $X=Y+Z$ and $Y \cap Z=\left\{ 0\right\}$.

Let $X$ be a normed linear space and let $Y \subset X$ be a finitedimensional linear subspace. Prove that there exists a vector $x_{0} \in X$ such that $d\left(x_{0}, Y\right)=1$.

Let $X$ be a normed linear space and $x, y$ two distinct points in $X$. Prove that there exists $T \in X^{*}$ for which $T x \neq T y$.

Let $V$ be the subspace of $L^{\infty }\left(0,1\right)$ defined by

Prove that there is $\varphi \in \left(L^{\infty }\left(0,1\right)\right)^{*}$ such that $\varphi (f)=\lim_{n} n \int_{[0,1 / n]} f(x) d x$ for every $f \in V$.

Prove that there exists a bounded linear functional LIM on $\ell^{\infty }$ with the following properties:

Let $X \neq 0$ and $Y$ be normed spaces for which $\mathcal{L}(X, Y)$ is a Banach space. Prove that $Y$ is also a Banach space.

Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ a linear operator such that $\varphi \circ T \in X^{*}$ for every $\varphi \in Y^{*}$. Prove that $T$ is bounded.

Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ an isometric isomorphism. Prove that there exists an isometric isomorphism $S: X^{*} \rightarrow Y^{*}$.

Prove Corollary 12.2.8.

Corollary 12.2.8: A subspace $Y$ of a Banach space $X$ is dense in $X$ if and only if every bounded linear functional that vanishes on $Y$ must vanish on $X$.

Let $X$ be a normed linear space, let $Y \subset X$ be a closed linear subspace, let $z \in X \backslash Y$, and let $V=\left\{ y+\lambda z: y \in Y, \lambda \in \mathbb {R}\right\}$. Prove that $V$ is a closed subspace of $X$.

Let $X$ be a normed linear space, let $Y \subset X$ be a linear subspace, and let $x_{0} \in X \backslash Y$. Prove that

Let $X, Y$ be normed spaces, let $X_{0}$ a closed subspace of $X$, let $a \in X \backslash X_{0}$, and let $b \in Y$. Prove that there is $T \in \mathcal{L}(X, Y)$ that vanishes on $X_{0}$ and such that $T a=b$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space, let $Y$ be a Banach space, and let $A: X \rightarrow Y$ be such that $T \circ A$ is integrable for every $T \in Y^{*}$. Define $L: Y^{*} \rightarrow \mathbb {R}$ by $L(T)=\int_{X} T \circ A d \mu$. Prove that $L \in Y^{* *}$.

Let $S$ be a subset of a normed linear space $X$. Prove that $S$ is bounded if and only if $T(S)$ is bounded for every $T \in X^{*}$.

Let $X$ be a normed linear space and $x_{0} \in X$. Prove that

Prove that every finite-dimensional normed linear space is reflexive.

Prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive.

Let $X$ be a reflexive normed space. Prove that if $Y$ is a closed subspace of $X$, then $Y$ is reflexive.

Let $X$ be a reflexive normed space and let $Y$ be a Banach space such that $Y^{*}$ is isometrically isomorphic to $X^{*}$. Prove that $Y$ is isometrically isomorphic to $X$.

Let $X$ be a reflexive normed space. Prove that every functional $T \in X^{*}$ attains its norm on $X$.

Let $X$ be a Banach space. If $\mathcal{M} \subset X$ and $\mathcal{N} \subset X^{*}$ are closed subspaces, let $\mathcal{M}^{0}=\left\{ T \in X^{*}: T \mid \mathcal{M}=0\right\}$ and $\mathcal{N}^{\perp }=\left\{ x \in X: T x=0 , \text{ for all } T \in \mathcal{N}\right\}$. Prove that:

Let $X$ be a normed space and $T, T_{1}, \ldots , T_{n} \in X^{*}$. Let $Y=\left\{ x \in X: T_{1}(x)=\cdots =T_{n}(x)=0\right\}$ and $S: Y \rightarrow \mathbb {R}$ be defined by $S(x)=T(x)$ on $Y$. If $\varphi$ is a bounded linear functional on $X$ that extends $S$ prove that there are $\alpha_{1}, \ldots , \alpha_{n} \in \mathbb {R}$ such that $\varphi =T+\alpha_{1} T_{1}+\cdots +\alpha_{n} T_{n}$.

Let $\mathcal{A}=\left\{ T \in \mathcal{L}\left(\ell^{2}\right): \left\langle T e_{1}, e_{1} \right\rangle -2 \left\langle T e_{1}, e_{2} \right\rangle =0\right\}$, and $\varphi$ a bounded linear functional on $\mathcal{A}$ defined by $\varphi (T)= \left\langle T e_{1}, e_{1} \right\rangle$. Prove that $\tilde{\varphi }: \mathcal{L}\left(\ell^{2}\right) \rightarrow \mathbb {R}$ defined by

is the unique Hahn-Banach extension of $\varphi$.