How to show if something is an inner product or not?
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I would appreciate some assistance in answering the following problems. We are moving so quickly through our advanced linear algebra material, I can't wrap my head around the key concepts. Thank you.
Let VV be the space of all continuously differentiable real valued functions on [a,b][a,b].
(i) Define
⟨f,g⟩=∫baf(t)g(t)dt+∫baf′(t)g′(t)dt.⟨f,g⟩=∫abf(t)g(t)dt+∫abf′(t)g′(t)dt.
Prove that ⟨,⟩⟨,⟩ is an inner product on VV.
(ii) Define that||f||=∫ba|f(t)|dt+∫ba|f′(t)|dt||f||=∫ab|f(t)|dt+∫ab|f′(t)|dt. Prove that this defines a norm on V.
Let VV be the space of all continuously differentiable real valued functions on [a,b][a,b].
(i) Define
⟨f,g⟩=∫baf(t)g(t)dt+∫baf′(t)g′(t)dt.⟨f,g⟩=∫abf(t)g(t)dt+∫abf′(t)g′(t)dt.
Prove that ⟨,⟩⟨,⟩ is an inner product on VV.
(ii) Define that||f||=∫ba|f(t)|dt+∫ba|f′(t)|dt||f||=∫ab|f(t)|dt+∫ab|f′(t)|dt. Prove that this defines a norm on V.
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