Bochner space

Type of topological space

In mathematics, Bochner spaces are a generalization of the concept of  L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not necessarily the space  R {\displaystyle \mathbb {R} } or  C {\displaystyle \mathbb {C} } of real or complex numbers.

The space  L p ( X ) {\displaystyle L^{p}(X)} consists of (equivalence classes of) all Bochner measurable functions  f {\displaystyle f} with values in the Banach space  X {\displaystyle X} whose norm ‖ f ‖ X {\displaystyle \|f\|_{X}} lies in the standard  L p {\displaystyle L^{p}} space. Thus, if  X {\displaystyle X} is the set of complex numbers, it is the standard Lebesgue  L p {\displaystyle L^{p}} space.

Almost all standard results on  L p {\displaystyle L^{p}} spaces do hold on Bochner spaces too; in particular, the Bochner spaces  L p ( X ) {\displaystyle L^{p}(X)} are Banach spaces for  1 ≤ p ≤ ∞ . {\displaystyle 1\leq p\leq \infty .}

Bochner spaces are named for the mathematician Salomon Bochner.

## Definition

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Given a measure space ( T , Σ ; μ ) , {\displaystyle (T,\Sigma ;\mu ),} a Banach space ( X , ‖ ⋅ ‖ X ) {\displaystyle \left(X,\|\,\cdot \,\|_{X}\right)} and  1 ≤ p ≤ ∞ , {\displaystyle 1\leq p\leq \infty ,} the Bochner space L p ( T ; X ) {\displaystyle L^{p}(T;X)} is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions  u : T → X {\displaystyle u:T\to X} such that the corresponding norm is finite:  ‖ u ‖ L p ( T ; X ) := ( ∫ T ‖ u ( t ) ‖ X p d μ ( t ) ) 1 / p < + ∞  for  1 ≤ p < ∞ , {\displaystyle \|u\|_{L^{p}(T;X)}:=\left(\int _{T}\|u(t)\|_{X}^{p}\,\mathrm {d} \mu (t)\right)^{1/p}<+\infty {\mbox{ for }}1\leq p<\infty ,} ‖ u ‖ L ∞ ( T ; X ) := e s s s u p t ∈ T ‖ u ( t ) ‖ X < + ∞ . {\displaystyle \|u\|_{L^{\infty }(T;X)}:=\mathrm {ess\,sup} _{t\in T}\|u(t)\|_{X}<+\infty .}

In other words, as is usual in the study of  L p {\displaystyle L^{p}} spaces,  L p ( T ; X ) {\displaystyle L^{p}(T;X)} is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a  μ {\displaystyle \mu } -measure zero subset of  T . {\displaystyle T.} As is also usual in the study of such spaces, it is usual to abuse notation and speak of a "function" in  L p ( T ; X ) {\displaystyle L^{p}(T;X)} rather than an equivalence class (which would be more technically correct).

## Applications

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Bochner spaces are often used in the functional analysis approach to the study of partial differential equations that depend on time, e.g. the heat equation: if the temperature  g ( t , x ) {\displaystyle g(t,x)} is a scalar function of time and space, one can write  ( f ( t ) ) ( x ) := g ( t , x ) {\displaystyle (f(t))(x):=g(t,x)} to make  f {\displaystyle f} a family  f ( t ) {\displaystyle f(t)} (parametrized by time) of functions of space, possibly in some Bochner space.

### Application to PDE theory

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Very often, the space  T {\displaystyle T} is an interval of time over which we wish to solve some partial differential equation, and  μ {\displaystyle \mu } will be one-dimensional Lebesgue measure. The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time. For example, in the solution of the heat equation on a region  Ω {\displaystyle \Omega } in  R n {\displaystyle \mathbb {R} ^{n}} and an interval of time  [ 0 , T ] , {\displaystyle [0,T],} one seeks solutions  u ∈ L 2 ( [ 0 , T ] ; H 0 1 ( Ω ) ) {\displaystyle u\in L^{2}\left([0,T];H_{0}^{1}(\Omega )\right)} with time derivative  ∂ u ∂ t ∈ L 2 ( [ 0 , T ] ; H − 1 ( Ω ) ) . {\displaystyle {\frac {\partial u}{\partial t}}\in L^{2}\left([0,T];H^{-1}(\Omega )\right).} Here  H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in  L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω);  H − 1 ( Ω ) {\displaystyle H^{-1}(\Omega )} denotes the dual space of  H 0 1 ( Ω ) . {\displaystyle H_{0}^{1}(\Omega ).}

(The "partial derivative" with respect to time  t {\displaystyle t} above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)

## See also

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* Bochner integral – Concept in mathematics
* Vector measure
* Vector-valued functions – Function valued in a vector space; typically a real or complex onePages displaying short descriptions of redirect targets
* Weakly measurable function

## References

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* Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.

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