Lambda abstraction

A lambda abstraction is another name for an anonymous function. It gets its name from the usual notation for writing it - for example, $\lambda x\to x^{2}$. (Note: some sources write it as $\lambda x.\ x^{2}$.)

In Haskell source code, the Greek letter lambda is replaced by a backslash character ('\') instead, since this is easier to type. (And requires only the basic 7-bit ASCII character set.) Similarly, the arrow is replaced with the must more ugly character sequence '->'. So, for example, the lambda abstraction above would be written in Haskell as

  \ x -> x * x

There is actually a whole mathematical theory devoted to expressing computation entirely using lambda abstractions - the lambda calculus. Most functional programming languages (including Haskell) are based upon some extension of this idea.

When a lambda abstraction is applied to a value - for instance, $(\lambda x\to x^{2})\ 7$ - the result of the expression is determined by replacing every occurrence of the parameter variable (in this case $x$) with the parameter value (in this case 7). This is an eta reduction.