What Does It Mean for a Function to Have a Continuous Extension

Mathematical method in functional analysis

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space X {\displaystyle X} by first defining a linear transformation T {\displaystyle {\mathsf {T}}} on a dense subset of X {\displaystyle X} and then extending T {\displaystyle {\mathsf {T}}} to the whole space via the theorem below. The resulting extension remains linear and bounded (thus continuous).

This procedure is known as continuous linear extension.

Theorem [edit]

Every bounded linear transformation T {\displaystyle {\mathsf {T}}} from a normed vector space X {\displaystyle X} to a complete, normed vector space Y {\displaystyle Y} can be uniquely extended to a bounded linear transformation T ~ {\displaystyle {\tilde {\mathsf {T}}}} from the completion of X {\displaystyle X} to Y . {\displaystyle Y.} In addition, the operator norm of T {\displaystyle {\mathsf {T}}} is c {\displaystyle c} if and only if the norm of T ~ {\displaystyle {\tilde {\mathsf {T}}}} is c . {\displaystyle c.}

This theorem is sometimes called the BLT theorem.

Application [edit]

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval [ a , b ] {\displaystyle [a,b]} is a function of the form: f r 1 1 [ a , x 1 ) + r 2 1 [ x 1 , x 2 ) + + r n 1 [ x n 1 , b ] {\displaystyle f\equiv r_{1}{\mathit {1}}_{[a,x_{1})}+r_{2}{\mathit {1}}_{[x_{1},x_{2})}+\cdots +r_{n}{\mathit {1}}_{[x_{n-1},b]}} where r 1 , , r n {\displaystyle r_{1},\ldots ,r_{n}} are real numbers, a = x 0 < x 1 < < x n 1 < x n = b , {\displaystyle a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b,} and 1 S {\displaystyle {\mathit {1}}_{S}} denotes the indicator function of the set S . {\displaystyle S.} The space of all step functions on [ a , b ] , {\displaystyle [a,b],} normed by the L {\displaystyle L^{\infty }} norm (see Lp space), is a normed vector space which we denote by S . {\displaystyle {\mathcal {S}}.} Define the integral of a step function by: I ( i = 1 n r i 1 [ x i 1 , x i ) ) = i = 1 n r i ( x i x i 1 ) . {\displaystyle {\mathsf {I}}\left(\sum _{i=1}^{n}r_{i}{\mathit {1}}_{[x_{i-1},x_{i})}\right)=\sum _{i=1}^{n}r_{i}(x_{i}-x_{i-1}).} I {\displaystyle {\mathsf {I}}} as a function is a bounded linear transformation from S {\displaystyle {\mathcal {S}}} into R . {\displaystyle \mathbb {R} .} [1]

Let P C {\displaystyle {\mathcal {PC}}} denote the space of bounded, piecewise continuous functions on [ a , b ] {\displaystyle [a,b]} that are continuous from the right, along with the L {\displaystyle L^{\infty }} norm. The space S {\displaystyle {\mathcal {S}}} is dense in P C , {\displaystyle {\mathcal {PC}},} so we can apply the BLT theorem to extend the linear transformation I {\displaystyle {\mathsf {I}}} to a bounded linear transformation I ~ {\displaystyle {\tilde {\mathsf {I}}}} from P C {\displaystyle {\mathcal {PC}}} to R . {\displaystyle \mathbb {R} .} This defines the Riemann integral of all functions in P C {\displaystyle {\mathcal {PC}}} ; for every f P C , {\displaystyle f\in {\mathcal {PC}},} a b f ( x ) d x = I ~ ( f ) . {\displaystyle \int _{a}^{b}f(x)dx={\tilde {\mathsf {I}}}(f).}

The Hahn–Banach theorem [edit]

The above theorem can be used to extend a bounded linear transformation T : S Y {\displaystyle T:S\to Y} to a bounded linear transformation from S ¯ = X {\displaystyle {\bar {S}}=X} to Y , {\displaystyle Y,} if S {\displaystyle S} is dense in X . {\displaystyle X.} If S {\displaystyle S} is not dense in X , {\displaystyle X,} then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.

See also [edit]

  • Continuous linear operator
  • Hahn–Banach theorem – Theorem on extension of bounded linear functionals
  • Vector-valued Hahn–Banach theorems

References [edit]

  • Reed, Michael; Barry Simon (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. San Diego: Academic Press. ISBN0-12-585050-6.

Footnotes [edit]

adamsprestriall.blogspot.com

Source: https://en.wikipedia.org/wiki/Continuous_linear_extension

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