Bounded Linear Operator

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1 Bounded Linear Operator and the Corresponding Space

Theorem 1.1 Let and be normed spaces. is a linear operator, then the following properties are equivalent:

(1) is continuous (in the sense of continuity for operators in metric spaces).

(2) is continuous at .

(3) is bounded.

Theorem 2.2 (The space of bounded linear operators) Let and be normed spaces. is a normed space with . If is Banach space, then is Banach space.

2 Baire Category Theorem and its Corollaries

2.1 Baire Category Theorem

Definition 2.1 (Nowhere dense set): Let a metric space, we say is nowhere dense if . (For any open set , .)

Theorem 2.1 (Baire category theorem): Any complete metric space is not mearge, i.e. can not be written as a countable union of nowhere dense sets.

We use this theorem in the following way: Let be nonempty complete metric space. Let be a sequence of closed subsets such that . Then there exists some such that

$$Int (X_{n_0}) \neq \varnothing$$

2.2 Four Corollaries

Definition 2.2 (Open mapping) We say is an open mapping if is open in whenever is open.

Theorem 2.2 (Open mapping theorem). Let and be Banach spaces and suppose that . If maps onto , then is an open mapping.

Corollary 2.3 (Bounded inverse theorem) Let and be Banach spaces and suppose that . If is bijective, then .

Definition 2.3 (Closed operator) Let and be normed spaces. Let defined on subspace of , be a linear operator.

(1) Assume that the graph , is closed in with norm (also a normed space), then is a closed operator.

(2) For any , if and (), then and is a closed operator.

Theorem 2.4 (Closed graph theorem) Let and be Banach spaces and a closed linear operator. Then is bounded (continuous).

Theorem 2.5 (Uniform boundedness theorem) Let be a Banach space, a normed linear space and let be subset of such that for any , such that,

$$\mathop{\Vert Tx\Vert}\limits_{T\in \mathcal F}<M_x$$

then there exists $M$ such that $\mathop{sup}\limits_{T\in \mathcal{F}}\Vert T\Vert<M$.

Corollary 2.6 Let $X$ be a Banach space, $Y$ be a normed space, and $T,\{T_n\}\in\mathcal L(X,Y)$. Assume $X_0\subseteq X$ is a dense subset. Then for any $x\in X$, $\mathop{lim}\limits_{n\rightarrow \infty}T_nx=Tx$ iff

$$\{||T_n||\} \text{ is bounded}; \mathop{lim}\limits_{n\rightarrow \infty}T_ny=Ty ,\forall y\in X_0$$

Proof See [2] Corollary 2.25.

Corollary 2.7 Let be Banach space, then is complete with respect to strong operator topology.

Proof. See [1] Corollary 6.2.3.

3 Hahn-Banach Theorem and its Consequences

3.1 Hahn-Banach Theorem

Theorem 3.1 (Zorn’s lemma) Let be a partially ordered set. If each totally ordered subset of has an upper bound, then has a maximal element.

Zorn’s Lemma is equivalent to Axiom of Choice: if is any collection of non-empty sets, then also is non-empty.

Definition 3.1 (Sublinear functional) Let be a linear space. The Mapping is called sublinear functional if

(1)

(2) .

Theorem 3.2 (Hahn-Banach theorem) be a proper linear subspace of a real linear space . Suppose that a sublinear functional defined on X, and a linear functional defined on such that for all . Then can be extended to a linear functional defined on such that for .

Proof. see [1].

3.2 Consequences of Hahn-Banach Theorem

Corollary 3.3 Let be a linear subspace. If is a bounded linear functional, then there exists that extends and such that

$$\Vert F\Vert_{X^*}=\Vert f\Vert_{X_0^*}$$

Corollary 3.4 Let be a linear subspace of a normed linear space and such that

$$d(x,x_0)=\mathop{inf}_{x_0\in X_0}||x-x_0||=d$$

then there exists such that,

$$||x^*||_{X^*}=1,x^*(x)=d \\x^*(x_0)=0,\forall x_0\in X_0$$

Corollary 3.5 Let be a normed linear space and , then there exists such that and .

Reference

[1] FUNCTIONAL ANALYSIS NOTES by Mr. Andrew Pinchuck, https://uomustansiriyah.edu.iq/media/lectures/9/9_2017_09_30!12_03_54_PM.pdf

[2] Functional Analysis Notes by He Jiazhi.