Banach space
Classical spaces

Basic examples of Banach spaces include: the Lp spaces L p {\displaystyle L^{p}} and their special cases, the sequence spaces ℓ p {\displaystyle \ell ^{p}} that consist of scalar sequences indexed by natural numbers N {\displaystyle \mathbb {N} }; among them, the space ℓ 1 {\displaystyle \ell ^{1}} of absolutely summable sequences and the space ℓ 2 {\displaystyle \ell ^{2}} of square summable sequences; the space c 0 {\displaystyle c_{0}} of sequences tending to zero and the space ℓ ∞ {\displaystyle \ell ^{\infty }} of bounded sequences; the space C (K) {\displaystyle C(K)} of continuous scalar functions on a compact Hausdorff space K, {\displaystyle K,} equipped with the max norm, ‖ f ‖ C (K) = max { | f (x) |: x ∈ K }, f ∈ C (K). {\displaystyle \|f\|_{C(K)}=\max\{|f(x)|:x\in K\},\quad f\in C(K).}

According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some C (K). {\displaystyle C(K).} For every separable Banach space X, {\displaystyle X,} there is a closed subspace M {\displaystyle M} of ℓ 1 {\displaystyle \ell ^{1}} such that X:= ℓ 1 / M. {\displaystyle X:=\ell ^{1}/M.}

Any Hilbert space serves as an example of a Banach space. A Hilbert space H {\displaystyle H} on K = R, C {\displaystyle \mathbb {K} =\mathbb {R},\mathbb {C} } is complete for a norm of the form ‖ x ‖ H = ⟨ x, x ⟩, {\displaystyle \|x\|_{H}={\sqrt {\langle x,x\rangle }},} where ⟨ ⋅, ⋅ ⟩: H × H → K {\displaystyle \langle \cdot,\cdot \rangle:H\times H\to \mathbb {K} } is the inner product, linear in its first argument that satisfies the following: ⟨ y, x ⟩ = ⟨ x, y ⟩ ¯, for all x, y ∈ H ⟨ x, x ⟩ ≥ 0, for all x ∈ H ⟨ x, x ⟩ = 0 if and only if x = 0. {\displaystyle {\begin{aligned}\langle y,x\rangle &={\overline {\langle x,y\rangle }},\quad {\text{ for all }}x,y\in H\\\langle x,x\rangle &\geq 0,\quad {\text{ for all }}x\in H\\\langle x,x\rangle =0{\text{ if and only if }}x&=0.\end{aligned}}}

For example, the space L 2 {\displaystyle L^{2}} is a Hilbert space.

The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to L p {\displaystyle L^{p}} spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.
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