Functional Analysis Formula Cheat Sheet for Spaces

Functional analysis formula cheat sheet covers normed spaces, operators, convergence, and Hilbert spaces, helping understand advanced mathematical concepts used in physics, engineering, and analysis.

Functional Analysis Formula Cheat Sheet

Topic	Formula	Description
Norm Definition	‖x‖ ≥ 0, ‖x‖=0⇔x=0	Length of vector
Scalar Rule	‖αx‖ =	α
Triangle Inequality	‖x+y‖ ≤ ‖x‖ + ‖y‖	Norm property
Normed Space	(X, ‖·‖)	Vector space with norm
Banach Space	Complete normed space	All Cauchy converge
Inner Product	⟨x,y⟩	General dot product
Induced Norm	‖x‖ = √⟨x,x⟩	From inner product
Cauchy-Schwarz		⟨x,y⟩
Parallelogram Law	‖x+y‖² + ‖x−y‖² = 2(‖x‖²+‖y‖²)	Norm identity
Hilbert Space	Complete inner product space	Key structure
Orthogonality	⟨x,y⟩ = 0	Perpendicular vectors
Orthonormal Set	⟨eᵢ,eⱼ⟩ = δᵢⱼ	Unit orthogonal
Projection	projᵧx = ⟨x,y⟩/‖y‖² · y	Closest vector
Gram-Schmidt	Orthonormalization	Basis creation
Bessel Inequality	Σ	⟨x,eᵢ⟩
Parseval Identity	Σ	⟨x,eᵢ⟩
Linear Operator	T(x+y)=T(x)+T(y)	Linear mapping
Bounded Operator	‖Tx‖ ≤ C‖x‖	Continuous operator
Operator Norm	‖T‖ = sup ‖Tx‖/‖x‖	Size measure
Adjoint Operator	⟨Tx,y⟩ = ⟨x,T*y⟩	Dual mapping
Self-Adjoint	T = T*	Symmetric operator
Unitary Operator	T*T = I	Preserves norm
Compact Operator	Maps bounded → compact	Special operator
Spectrum	σ(T)	Eigenvalue set
Spectral Radius	r(T) = max	λ
Hahn-Banach	Extend functionals	Extension theorem
Open Mapping	Onto linear map is open	Key theorem
Closed Graph	Closed graph ⇒ bounded	Operator property
Uniform Boundedness	sup‖Tₙx‖ bounded	Banach-Steinhaus
Weak Convergence	⟨xₙ,f⟩ → ⟨x,f⟩	Weak limit
Weak* Convergence	Dual space convergence	Weak star
Reflexive Space	X ≅ X**	Double dual
Lᵖ Space	‖f‖ₚ = (∫	f
L¹ Norm	∫	f
L² Norm	(∫	f
L∞ Norm	sup	f
Holder Inequality	∫	fg
Minkowski Inequality	‖f+g‖ₚ ≤ ‖f‖ₚ + ‖g‖ₚ	Triangle inequality
Banach Fixed Point	Contraction ⇒ unique fixed point	Existence
Sobolev Space	Functions with weak derivatives	Advanced space

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Conclusion The functional analysis examines spaces of functions and their characteristics. It is an amalgamation of algebra, analysis and topology. The functional analysis is common in physics, engineering and mathematics. It is concerned with such notions as normed spaces, Banach spaces, and operators. Functional analysis is a learning that enhances abstract thinking and analytical skills. It has applications in quantum mechanics and signal processing. Infinite-dimensional spaces can be studied with the use of functional analysis. It offers the aid of advanced mathematical problems. In contemporary mathematics, it is a major topic. All in all, the study of functional analysis is profound and significant. In summary, functional analysis studies function spaces and operators. It is essential for advanced mathematics and has many applications in science and engineering.