Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

## D. Trace-Class/Compact Duality

Proposition III.35

Let $$$Φ : H\times H \to \mathbb{C}$$$ be sesquilinear (conjugate-bilinear) and suppose there exists $$$M\geq 0$$$ such that $$$\mathopen{}\left\lvert{}Φ\mathopen{}\left( x, y\right)\mathclose{}\right\rvert\mathclose{}\leq M\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$. Then there exists $$$S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$ such that $$$Φ\mathopen{}\left( x, y\right)\mathclose{}= \mathopen{}\left\langle{}S\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}$$$.

Proof. For $$$y\in H$$$, consider $$$x\mapsto Φ\mathopen{}\left( x, y\right)\mathclose{}$$$. This is a bounded linear functional on $$$H$$$ of norm at most $$$M\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$. Get $$$R\mathopen{}\left( y\right)\mathclose{}\in H$$$ such that $$$Φ\mathopen{}\left( x, y\right)\mathclose{}= \mathopen{}\left\langle{}x, R\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}$$$ with $$$\mathopen{}\left\lVert{}R\mathopen{}\left( y\right)\mathclose{}\right\rVert\mathclose{}\leq M\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$. By uniqueness of $$$R\mathopen{}\left( y\right)\mathclose{}$$$ for any given $$$y$$$, we get that $$$R : H \to H$$$ is linear, so $$$R\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$ with $$$\mathopen{}\left\lVert{}R\right\rVert\mathclose{}\leq M$$$. Let $$$S= R^{*}$$$.

Proposition III.36

For every $$$φ\in {\mathcal{K}\mathopen{}\left( H\right)\mathclose{}}^{*}$$$ (the dual space of $$$\mathcal{K}\mathopen{}\left( H\right)\mathclose{}$$$), there exists $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$ such that $$$φ\mathopen{}\left( K\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( TK\right)\mathclose{}$$$ for all $$$K\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{}$$$. This $$$T$$$ is unique and $$$\mathopen{}\left\lVert{}φ\right\rVert\mathclose{}= \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$.

Proof. Define $$$Φ : H\times H \to \mathbb{C}$$$ by $$$Φ\mathopen{}\left( x, y\right)\mathclose{}= φ\mathopen{}\left( x\otimes y\right)\mathclose{}$$$. Then $$$Φ$$$ satisfies the hypothesis of Proposition III.35 because $$$\mathopen{}\left\lvert{}Φ\mathopen{}\left( x, y\right)\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\mathopen{}\left\lVert{}x\otimes y\right\rVert\mathclose{}= \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$. Thus there exists $$$T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$ such that $$$φ\mathopen{}\left( x\otimes y\right)\mathclose{}= \mathopen{}\left\langle{}T\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}$$$. Let $$$\mathopen{}\left({e}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$ be an orthonormal basis for $$$H$$$. Polarize $$$T= V\mathopen{}\left\lvert{}T\right\rvert\mathclose{}$$$. Let $$${P}_{n}= \sum_{j=1}^{n}{} {e}_{j}\otimes {e}_{j}$$$ (the projection on $$$\operatorname{span}\mathopen{}\left( {e}_{1}, \dotsc, {e}_{n}\right)\mathclose{}$$$). Then $$${P}_{n} V^{*}= \sum_{j=1}^{n}{} {e}_{j}\otimes V{e}_{j}$$$, $$$\mathopen{}\left\lVert{}{P}_{n} V^{*}\right\rVert\mathclose{}\leq 1$$$, and $$${P}_{n} V^{*}\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$. So, $$\mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\geq \mathopen{}\left\lvert{}φ\mathopen{}\left( {P}_{n} V^{*}\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}φ\mathopen{}\left( \sum_{j=1}^{n}{} {e}_{j}\otimes V\mathopen{}\left( {e}_{j}\right)\mathclose{} \right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\sum_{j=1}^{n}{} \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{j}\right)\mathclose{}, V\mathopen{}\left( {e}_{j}\right)\mathclose{}\right\rangle\mathclose{} \right\rvert\mathclose{}= \mathopen{}\left\lvert{}\sum_{j=1}^{n}{} \mathopen{}\left\langle{}V\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{j}\right)\mathclose{}\right)\mathclose{}, V\mathopen{}\left( {e}_{j}\right)\mathclose{}\right\rangle\mathclose{} \right\rvert\mathclose{}= \mathopen{}\left\lvert{}\sum_{j=1}^{n}{} \mathopen{}\left\langle{} V^{*}\mathopen{}\left( V\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{j}\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, {e}_{j}\right\rangle\mathclose{} \right\rvert\mathclose{}= \sum_{j=1}^{n}{} \mathopen{}\left\langle{}\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{j}\right)\mathclose{}, {e}_{j}\right\rangle\mathclose{} \text{.}$$ Let $$$n$$$ approach $$$\infty$$$. We find $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$ and $$$\operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\leq \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}$$$. Next, $$$φ\mathopen{}\left( x\otimes y\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( Tx\otimes y\right)\mathclose{}$$$ for all $$$x$$$ and $$$y$$$. This implies $$$φ\mathopen{}\left( F\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( TF\right)\mathclose{}$$$ for all $$$F\in \mathcal{F}\mathopen{}\left( H\right)\mathclose{}$$$. For general $$$K\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{}$$$, get $$$\mathopen{}\left({F}_{n}\right)\mathclose{}\in \mathcal{F}\mathopen{}\left( H\right)\mathclose{}$$$ such that $$${F}_{n} \to K$$$ (in norm). Then $$$φ\mathopen{}\left( {F}_{n}\right)\mathclose{} \to φ\mathopen{}\left( K\right)\mathclose{}$$$, and, by Remark III.28, $$$\mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( T\mathopen{}\left({F}_{n}-K\right)\mathclose{}\right)\mathclose{}\right\rvert\mathclose{}\leq \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\mathopen{}\left\lVert{}{F}_{n}-K\right\rVert\mathclose{} \to 0$$$, i.e. $$$\operatorname{Tr}\mathopen{}\left( T{F}_{n}\right)\mathclose{} \to \operatorname{Tr}\mathopen{}\left( TK\right)\mathclose{}$$$. So $$$φ\mathopen{}\left( K\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( TK\right)\mathclose{}$$$ for all $$$K\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{}$$$. So, $$$\mathopen{}\left\lvert{}φ\mathopen{}\left( K\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( T\mathopen{}\left( K\right)\mathclose{}\right)\mathclose{}\right\rvert\mathclose{}\leq \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\mathopen{}\left\lVert{}K\right\rVert\mathclose{}$$$ for all $$$K$$$, and so $$$\mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\leq \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$, and $$$\mathopen{}\left\lVert{}φ\right\rVert\mathclose{}= \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$.

Proposition III.37

Define $$$\mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}_{\operatorname{Tr}}$$$ on $$$\mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$ by $$$\mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}= \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$.

1. $$$\mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}_{\operatorname{Tr}}$$$ is a complete norm on $$$\mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$.
2. $$$\mathopen{}\left\lVert{}x\otimes y\right\rVert\mathclose{}_{\operatorname{Tr}}= \mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$.
3. With respect to $$$\mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}_{\operatorname{Tr}}$$$, $$$\overline{ \mathcal{F}\mathopen{}\left( H\right)\mathclose{} }= \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$.
4. $$$\mathopen{}\left\lVert{}T\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}$$$.
5. $$$\mathopen{}\left\lVert{}ST\right\rVert\mathclose{}_{\operatorname{Tr}}\leq \mathopen{}\left\lVert{}S\right\rVert\mathclose{}\mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}$$$ for all $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$, $$$S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$.

Proof.

1. $$$\mathcal{K}\mathopen{}\left( H\right)\mathclose{}^{*}$$$, like any dual space, is a Banach space.
2. We have $$\mathopen{}\left(x\otimes y\right)\mathclose{} ^{*}\mathopen{}\left(x\otimes y\right)\mathclose{}= \mathopen{}\left(y\otimes x\right)\mathclose{}\mathopen{}\left(x\otimes y\right)\mathclose{}= \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}y\otimes y= {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}^{2}\frac{y}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}\otimes \frac{y}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}= {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}^{2}P \text{}$$ where $$$P$$$ is the projection on $$$\mathbb{C}y$$$. So, $$$\mathopen{}\left\lvert{}x\otimes y\right\rvert\mathclose{}= \mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}\frac{y}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}\otimes \frac{y}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}$$$. Thus $$$\operatorname{Tr}\mathopen{}\left( x\otimes y\right)\mathclose{}= \mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$.
3. Take $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$. Let $$$\mathopen{}\left({e}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$ be an orthonormal basis diagonalizing $$$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}$$$. Then $$$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}= \sum{} {λ}_{n}{e}_{n}\otimes {e}_{n}$$$, with $$${λ}_{n}\geq 0$$$ and $$$\sum{} {λ}_{n} \lt \infty$$$. Write $$${P}_{j}= \sum_{n=1}^{j}{} {e}_{n}\otimes {e}_{n}$$$ (which is the projection on $$$\operatorname{span}\mathopen{}\left( {e}_{1}, {e}_{2}, \dotsc, {e}_{j}\right)\mathclose{}$$$). Then $$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left(1-{P}_{j}\right)\mathclose{}= \sum_{n=j+1}^{\infty}{} {λ}_{n}{e}_{n}\otimes {e}_{n}$$ so $$$\mathopen{}\left\lVert{}\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left(1-{P}_{j}\right)\mathclose{}\right\rVert\mathclose{}_{\operatorname{Tr}}= \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left(1-{P}_{j}\right)\mathclose{}\right)\mathclose{}= \sum_{n=j+1}^{\infty}{} {λ}_{n} \to 0$$$. Finally, the polar decomposition $$$T= V\mathopen{}\left\lvert{}T\right\rvert\mathclose{}$$$ gives $$\mathopen{}\left\lVert{}T\mathopen{}\left(1-{P}_{j}\right)\mathclose{}\right\rVert\mathclose{}_{\operatorname{Tr}}\leq \mathopen{}\left\lVert{}V\right\rVert\mathclose{}\operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left(1-{P}_{j}\right)\mathclose{}\right)\mathclose{} \to 0 \text{.}$$ Thus $$$\mathopen{}\left\lVert{}T-T{P}_{j}\right\rVert\mathclose{} \to 0$$$.
4. $$$\mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}= \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$, which is the sum of the eigenvalues of $$$T$$$, and is therefore at least $$$ρ\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$ (equal to the largest eigenvalue of $$$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}$$$). Thus $$\mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}\geq ρ\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}= \mathopen{}\left\lVert{}\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right\rVert\mathclose{}= {\mathopen{}\left\lVert{}{\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{2}\right\rVert\mathclose{}}^{\frac{1}{2}}= {\mathopen{}\left\lVert{} T^{*}T\right\rVert\mathclose{}}^{\frac{1}{2}}= \mathopen{}\left\lVert{}T\right\rVert\mathclose{} \text{.}$$
5. $$$\mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( STK\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( TKS\right)\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}\mathopen{}\left\lVert{}KS\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{}_{\operatorname{Tr}}\mathopen{}\left\lVert{}S\right\rVert\mathclose{}\mathopen{}\left\lVert{}K\right\rVert\mathclose{}$$$ for all $$$K\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{}$$$.

Proposition III.38

For every $$$f\in { \mathopen{}\left(\mathcal{T}\mathopen{}\left( H\right)\mathclose{}, \mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}_{\operatorname{Tr}}\right)\mathclose{} }^{*}$$$, there exists a unique $$$S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$ such that $$$f\mathopen{}\left( T\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( ST\right)\mathclose{}$$$ for all $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$. Furthermore, $$$\mathopen{}\left\lVert{}f\right\rVert\mathclose{}= \mathopen{}\left\lVert{}S\right\rVert\mathclose{}$$$.

Proof. Consider the conjugate bilinear (sesquilinear) form $$$\mathopen{}\left(x, y\right)\mathclose{}\mapsto f\mathopen{}\left( x\otimes y\right)\mathclose{}$$$. Since $$$\mathopen{}\left\lvert{}f\mathopen{}\left( x\otimes y\right)\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}f\right\rVert\mathclose{}\mathopen{}\left\lVert{}x\otimes y\right\rVert\mathclose{}_{\operatorname{Tr}}= \mathopen{}\left\lVert{}f\right\rVert\mathclose{}\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}$$$, by Proposition III.35 there exists $$$S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$ such that $$$f\mathopen{}\left( x\otimes y\right)\mathclose{}= \mathopen{}\left\langle{}S\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}$$$ and $$$\mathopen{}\left\lVert{}S\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}f\right\rVert\mathclose{}$$$. Thus $$$f\mathopen{}\left( x\otimes y\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( S\mathopen{}\left( x\otimes y\right)\mathclose{}\right)\mathclose{}$$$ for all $$$x$$$ and $$$y$$$ in $$$H$$$. So $$$f\mathopen{}\left( F\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( SF\right)\mathclose{}$$$ for all $$$F\in \mathcal{F}\mathopen{}\left( H\right)\mathclose{}$$$. Then $$$f\mathopen{}\left( T\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( ST\right)\mathclose{}$$$ by Proposition III.37 for all $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$. Further $$$\mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( S\mathopen{}\left( T\right)\mathclose{}\right)\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}S\right\rVert\mathclose{}\operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}$$$ so $$$\mathopen{}\left\lVert{}f\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}S\right\rVert\mathclose{}$$$.

Remark III.39

Thus $$$\mathopen{}\left(\mathcal{L}\mathopen{}\left( H\right)\mathclose{}, \mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}\right)\mathclose{}\simeq { \mathopen{}\left(\mathcal{T}\mathopen{}\left( H\right)\mathclose{}, \mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}_{\operatorname{Tr}}\right)\mathclose{} }^{*}$$$, where $$$\mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{}$$$ is the operator norm. The resulting $$$\mathop{\mathrm{w}^*}$$$-topology is called the ultraweak topology on $$$\mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$. $$${S}_{α} \mathbin{\overset{u}{\to}}S$$$ if and only if $$$\operatorname{Tr}\mathopen{}\left( {S}_{α}\mathopen{}\left( T\right)\mathclose{}\right)\mathclose{} \to \operatorname{Tr}\mathopen{}\left( S\mathopen{}\left( T\right)\mathclose{}\right)\mathclose{}$$$ for all $$$T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}$$$ if and only if $$$\sum_{j}{} \mathopen{}\left\langle{}{S}_{α}\mathopen{}\left( {x}_{j}\right)\mathclose{}, {y}_{j}\right\rangle\mathclose{} \to \sum_{j}{} \mathopen{}\left\langle{}{S}_{j}, {y}_{j}\right\rangle\mathclose{}$$$ for all $$$\mathopen{}\left({x}_{j}\right)\mathclose{}$$$ and $$$\mathopen{}\left({y}_{j}\right)\mathclose{}$$$ in $$$H$$$ with the sum $$$\sum{} \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \lt \infty$$$. If any of these conditions hold, we then have $$$\mathopen{}\left\langle{}{S}_{α}\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{} \to \mathopen{}\left\langle{}S\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}$$$ (i.e. $$${S}_{α} \mathbin{\overset{\mathop{\mathrm{w}^*}}{\to}}S$$$; weakly).

Remark III.40

The weak and ultraweak topologies coincide on bounded sets. And, by Alaoglu's Theorem, the unit ball of $$$\mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$ is ultraweakly compact (hence weakly compact).