Continuous functions on a compact Hausdorff space

Banach Algebras

Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019

Eberhard Malkowsky, Vladimir Rakočević

Examples of C∗–algebras are the algebra of all complex numbers C with the involution λ→λ¯, the algebra of continuous functions of a compact Hausdorff space (see Example 6.3.3) with the involution f→f∗ and f∗(t)=f(t)¯, and the algebra of all bounded linear operators B(X) on a Hilbert space X with the involution A→A∗, where A∗ is the Hilbert adjoint operator of A.

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Published in Hugo D. Junghenn, Principles of Analysis, 2018

Hugo D. Junghenn

In this section we prove that weak compactness and weak sequential compactness in a Banach space are equivalent. We first prove the analogous result for the pointwise topology on the space of continuous functions on a compact Hausdorff space.

General system of cubic–quartic functional equations in quasi-β-normed spaces

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Published in International Journal of General Systems, 2022

Abasalt Bodaghi

Some concrete and basic examples of -algebras are as follows. The set with the usual addition and product with one of the following involutions: The set , the space of continuous functions on a compact Hausdorff space X equipped with the pointwise addition and product of functions and scalar multiplication by constants. Moreover, the norm defined by and the involution is , for all .The prototypical example of a -algebra is the algebra of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here, denotes the adjoint operator of the operator . In fact, every -algebra , is -isomorphic to a norm-closed adjoint closed subalgebra of for a suitable Hilbert space H; this is the content of the Gelfand–Naimark theorem.

Continuous functions on a compact Hausdorff space

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Banach Algebras

Miscellaneous Topics

General system of cubic–quartic functional equations in quasi-β-normed spaces