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Theorem

Every bounded linear transformation L {\displaystyle L} 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 L ^ {\displaystyle {\widehat {L}}} from the completion of X {\displaystyle X} to Y . {\displaystyle Y.} In addition, the operator norm of L {\displaystyle L} is c {\displaystyle c} if and only if the norm of L ^ {\displaystyle {\widehat {L}}} is c . {\displaystyle c.}

This theorem is sometimes called the BLT theorem.

Application

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}\mathbf {1} _{[a,x_{1})}+r_{2}\mathbf {1} _{[x_{1},x_{2})}+\cdots +r_{n}\mathbf {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 \mathbf {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 I\left(\sum _{i=1}^{n}r_{i}\mathbf {1} _{[x_{i-1},x_{i})}\right)=\sum _{i=1}^{n}r_{i}(x_{i}-x_{i-1}).} I {\displaystyle 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 I} to a bounded linear transformation I ^ {\displaystyle {\widehat {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={\widehat {I}}(f).}

The Hahn–Banach theorem

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

• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Continuous linear operator
• Densely defined operator – Function that is defined almost everywhere (mathematics)
• Hahn–Banach theorem – Theorem on extension of bounded linear functionals
• Linear extension (linear algebra) – Mathematical function, in linear algebraPages displaying short descriptions of redirect targets
• Partial function – Function whose actual domain of definition may be smaller than its apparent domain
• Vector-valued Hahn–Banach theorems

• Reed, Michael; Barry Simon (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. San Diego: Academic Press. ISBN 0-12-585050-6.

References

1. Here, R {\displaystyle \mathbb {R} } is also a normed vector space; R {\displaystyle \mathbb {R} } is a vector space because it satisfies all of the vector space axioms and is normed by the absolute value function. /wiki/Vector_space#Formal_definition ↩