Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 32: Duals

Duals

Let $𝔽=ℝ$ of $𝔽=ℂ$ and let $V$ be a normed vector space over $𝔽\text{.}$

A linear functional on $V$ is a linear operator $\phi :V\to 𝔽\text{.}$ The dual of $V$ is $$V^{*}=B(V,𝔽)=\{\phi :V\to 𝔽\hspace{0.17em}|\hspace{0.17em}\phi \hspace{0.17em}\text{is linear and}\hspace{0.17em}\Vert \phi \Vert <\infty \}\text{.}$$

Adjoints

Let $V$ and $W$ be normed vector spaces over $𝔽\text{.}$

Let $T:V\to W$ be a bounded linear operator.
The adjoint of $T$ is the function $T^{*}:W^{*}\to V^{*}$ given by $\left(T\psi \right)\left(v\right)=(\psi \circ T)\left(v\right)\text{.}$ $$\begin{array}{ccc}V& \stackrel{T}{⟶}& W\\ & & \downarrow \psi \\ & & 𝔽\end{array}$$

HW: Show that $T^{*}:W^{*}\to V^{*}$ is a linear operator.

HW: Show that $\Vert T^{*}\Vert =\Vert T\Vert \text{.}$

HW: Show that $$\begin{array}{cccc}\text{ev}:& V& ⟶& V^{**}\\ & x& ⟼& {\text{ev}}_x:& V& ⟶& 𝔽\\ & & & & \phi & ⟼& \phi \left(x\right)\end{array}$$ is an injective linear transformation and $\Vert x\Vert =\Vert {\text{ev}}_x\Vert \text{.}$

A normed vector space $V$ is reflexive if $\text{ev}:V\to V^{**}$ is a bijection.

HW: Show that if $V$ and $W$ are reflexive then $T^{**}=T\text{.}$

HW: Let $p\in {ℝ}_{>1}$ and let $q\in {ℝ}_{>1}$ be given by $\frac{1}{p}+\frac{1}{q}=1\text{.}$ Show that ${\left({\ell }^p\right)}^{*}={\ell }^q\text{.}$

HW:

(a)	Show that if $p\in {ℝ}_{>1}$ then ${\ell }^p$ is reflexive.
(b)	Show that ${\ell }^1$ is not reflexive.
(c)	Show that ${\ell }^{\infty }$ is not reflexive.

HW:

(a)	Show that ${\left({\ell }^1\right)}^{*}={\ell }^{\infty }\text{.}$
(b)	Show that ${\left({\ell }^{\infty }\right)}^{*}\ne {\ell }^1\text{.}$

(Riesz representation Theorem) Let $H$ be a Hilbert space. Then $$\begin{array}{ccc}H& ⟶& H^{*}\\ a& ⟼& {\phi }_a:& H& ⟶& 𝔽\\ & & & x& ⟼& ⟨x,a⟩\end{array}$$ is a vector space isomorphism.

HW: Let $(H_1,{⟨\hspace{0.17em}⟩}_{H_1})$ and $(H_2,{⟨\hspace{0.17em}⟩}_{H_2})$ be Hilbert spaces. Let $T:H_1\to H_2$ be a bounded linear operator. Show that the adjoint of $T$ is the function $T^{*}:H_2\to H_1$ given by ${⟨T^{*}y,x⟩}_{H_1}={⟨y,Tx⟩}_{H_2}\text{.}$

Finite dimensional vector spaces

Let $V$ and $W$ be finite dimensional vector spaces over $𝔽\text{.}$ Let $\{v_1,v_2,\dots ,v_n\}$ be a basis of $V$ and let $\{w_1,w_2,\dots ,w_m\}$ be a basis of $W\text{.}$

The dual basis to $\{v_1,v_2,\dots ,v_n\}$ is the basis $\{v^1,v^2,\dots ,v^n\}$ of $V^{*}$ given by $$v^i\left(v_j\right)=\{\begin{array}{ll}1,& \text{if}\hspace{0.17em}i=j,\\ 0,& \text{if}\hspace{0.17em}i\ne j,\end{array}\hspace{0.17em}i,j\in \{1,2,\dots ,n\}\text{.}$$ Let $\{w^1,w^2,\dots ,w^m\}$ be the dual basis to $\{w_1,w_2,\dots ,w_m\}\text{.}$ $\{w^1,w^2,\dots ,w^m\}$ is a basis of $W^{*}\text{.}$

HW: Let $T:V\to W$ be a linear operator and let $T_{ij}\in 𝔽$ be given by $$T{v}_i=\sum _{j=1}^m{T}_{ji}{w}_j\text{.}$$ Show that $$T^{*}{w}^j=\sum _{i=1}^n{T}_{ji}{v}^i$$ by evaluating each side at $v_k\text{.}$

HW: If $⟨,⟩$ is an inner product on $V$ and $\{v_1,v_2,\dots ,v_n\}$ is an orthonormal basis of $V$ with respect to $⟨,⟩$ and $$T:V⟶V$$ is a linear operator and $$T{v}_i=\sum _{j=1}^n{T}_{ji}{v}_j\text{then}{T}^{*}{v}_i=\sum _{i=1}^n{\bar{T}}_{ij}{v}_j\text{.}$$

Notes and References

These are a typed copy of Lecture 32 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 19, 2014.

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