Linear transformation

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.

Jun Kang
Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.