Functional Analysis 10 -- Direct Sum 1

This is course note from Siran Li-Shanghai Jiao Tong University.

Algebraically, for \(X\) as v.s. and subspace \(Y,Z \le X.\) We have another subspace \(Y + Z :={y+z: y \in Y, z \in Y}.\) Then there is a linear surjection \(T\). \[ T: Y \times Z \rightarrow Y+Z \quad (y,z) \rightarrow y+z \] \(T\) is injective iff \(Y \cap Z = \{0\}\). In this case, we write \(X_0 = Y \oplus Z\) (Algebraically Sum)

If \(X\) is a NVS, then \(T \in B(Y\times Z, Y+Z).\) Importantly, even if \(X_0 = Y \oplus Z,\) we don't know if \(T^{-1}\) is bounded.

Definition: Topological Direct Sum

If \(T: Y\times Z \rightarrow Y \oplus Z,\) is a NVS isomorphism, say \(Y \oplus Z\) is a topological direct sum.

Definition:

For \(Y \le X\), say \(Y\) is (algebraically/topologically) complemented in \(X\) iff \(\exists Z \le X,\) s.t. \(X = Y \oplus Z\)(algebraically/topologically).

Examples: \(l^2 \cong l^2 \oplus ... \oplus l^2\) finite times.

Proposition:

Every subspace is algebraically complemented. pf: Zorn's lemma.

Proposition:

\(Y \le X\) is algebraically complemented \(\Leftrightarrow\) \(\exists\) projection \(I: X \rightarrow X\) s.t. \(ran(I) = Y.\) Projection means \(I\) is linear and \(I^2 = I.\)

proof:([[TODO]])

Theorem:

Let \(X\) be an NVS, \(Y,Z \le X\) s.t. \(X = Y \oplus Z.\) Let \(P: X\rightarrow X\) be the projection onto \(Y.\) Then \(X = Y \oplus Z\) is a topologically direct sum iff \(P\) is bounded.

proof:([[TODO]])

Corollary: If \(X\) is a Banach space, \(Y, Z\) are closed subspaces and \(X = Y\oplus Z\)(alg.) Then \(X = Y \oplus Z\)(top.)

Rmk: Known that \(Y \le X\)(even closed), it is unknown whether \(Y\) is topologically complemented. Eg: \(c_0\) is not complemented in \(l^{\infty}.\)(Phillips 1940-50)

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