Summary of Key Theorems

Theorem	Statement
Hahn-Banach	Bounded functionals extend preserving norm
Open Mapping	Surjective bounded operator between Banach spaces is open
Closed Graph	A linear operator between Banach spaces is bounded iff its graph is closed
Uniform Boundedness	Pointwise-bounded family of operators is uniformly bounded
Riesz Representation	Every functional on a Hilbert space is given by inner product
Spectral Theorem	Compact self-adjoint operators have orthonormal eigenbasis
Fredholm Alternative	For compact $T$ and $\lambda \neq 0$ : either $\lambda I - T$ is invertible or has nontrivial kernel
Banach-Alaoglu	Closed unit ball of $X^$ is weak-compact

:::caution Common Pitfall The Closed Graph Theorem requires both the domain and codomain to be Banach spaces. A linear operator $T : C[0,1] \to C[0,1]$ defined by $Tf = f"$ (where defined) has a closed graph in $C[0,1] \times C[0,1]$ , but $C^1[0,1]$ is not complete under the $C[0,1]$ norm. The theorem does not apply, and $T$ is indeed unbounded.

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