For topological spaces X and Y, let C(X, Y) denote the collection of all continuous functions from X...

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For topological spaces X and Y, let C(X, Y) denote the collection of all continuous functions from X to Y. We will distinguish two special collections: C(X) will be used to denote C(X, R) and C*(X) will denote the set of all bounded functions from C(X). We can define addition, multiplication and scalar multiplication of functions in C(X) pointwise :
(f + g)(x) = f(x) + g(x),
(f ·g)(x) = f(x) ·g(x),
(a ·f)(x) = a ·f(x), for a ?R.
1. If f and g belong to C(X), then so do f + g, f ·g and a ·f, for a ?R. If, in addition,f and g are bounded, then so are f + g, f ·g and a ·f
2. C(X) and C*(X) are algebras over the real numbers. (Consult any book on abstract algebra for the definition of an algebra.)
3. C*(X) is a normed linear space (2J) with the operations of addition and scalar multiplication given above and the norm || f|| = sup _x_?_X lf(x)l.

Robert · Accepted Answer

1. Let xn → x0, as f and g are continuous, we have
lim
xn→x0
f(xn) = f(x0) and lim
xn→x0
g(xn) = g(x0)
Now by definition of f + g, f.g and a.f , we have
lim
xn→x0
(f + g)(xn) = lim
xn→x0
(f(xn) + g(xn)) = f(x0) + g(x0) = (f + g)(x0)
lim
xn→x0
(f.g)(xn) = lim
xn→x0
(f(xn).

For topological spaces X and Y, let C(X, Y) denote the collection of all continuous functions from X to Y . We will distinguish two special collections: C(X) will be used to denote C(X, R) and C*(X)...

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