Let C([0,T];C(¯U)) denote the set of all continuous functions u:[0,T]→C(¯U) with
‖u‖C([0,T];C(¯U)):=max0⩽t⩽T‖u(t)‖<∞
Prove that C([0,T];C(¯U))=C([0,T]ׯU)
I am skeptical this is even true. I feel like we could apply a theorem from topology regarding the product space, but am having little success. Not really sure how to approach such a problem. Are there any counter examples that disprove the above? Any help would be much appreciated.
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