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We introduce kernel-summability methods in Banach spaces using thevector-valued integrals and prove an analogue of the Silverman-Toeplitz Theoremfor regular kernel-summability methods. We also show that if $X$ is a Banachspace and one kernel-summability method is included in anotherkernel-summability method for scalar-valued functions, then the first method isincluded in the second method, for $X$-valued functions. This extends aprevious result from Javad Mashreghi, Thomas Ransford and the author. We thenapply these abstract results to the summability of Taylor series of functionsin a Banach space of holomorphic functions on the unit disk.

Kernel-Summability Methods and the Silverman-Toeplitz Theorem