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

This theorem is sometimes called the BLT theorem.

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval ${\displaystyle [a,b]}$  is a function of the form: ${\displaystyle f\equiv r_{1}\mathbf {1} _{[a,x_{1})}+r_{2}\mathbf {1} _{[x_{1},x_{2})}+\cdots +r_{n}\mathbf {1} _{[x_{n-1},b]}}$  where ${\displaystyle r_{1},\ldots ,r_{n}}$  are real numbers, ${\displaystyle a=x_{0}<x_{1}<\ldots <x_{n-1}<x_{n}=b,}$  and ${\displaystyle \mathbf {1} _{S}}$  denotes the indicator function of the set ${\displaystyle S.}$  The space of all step functions on ${\displaystyle [a,b],}$  normed by the ${\displaystyle L^{\infty }}$  norm (see Lp space), is a normed vector space which we denote by ${\displaystyle {\mathcal {S}}.}$  Define the integral of a step function by:

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

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

• 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.