A norm of linear transformation \(\Lambda : X \rightarrow Y\) is defined by \(\| \Lambda \| = \sup \{ \| \Lambda(x) \|:x \in X, \|x\| \le 1 \}\). We can give a norm to a space or a set. A norm determines the size of a vector in the function space. The way of measure the size of a vector gives important properties of the space like boundness, completeness or orthogonality. We can also select a set (\(L^{1}(\mu)\)) to have a certain properties like boundness and completeness in a specific norm like \(\| * \|_2\) and so on select \(C(X)\) for \(L^{\infty}\).
The linear transformation \(\Lambda\) can compose a normed space \(\{\Lambda\}\). The space of linear transformation can have properties like boundness, completeness. The Banach-Steinhaus theorem states the limit of linear transformation between Banach spaces which a set of bounded linear transformation, \(\{\Lambda_{\alpha}\}\), \(\alpha \in A\) with Banach space in both domain and codomain has either bounded norm or unbounded supremum \(\| \Lambda_{\alpha} \|\) for all some dense \(G_\delta\) in domain. The limit of linear transformation space can be applied in convergence of differentiation and integration of simple function and Fourier transformation because those are series of linear transformation.
