Which of the following is an inner product on vector space of all

real valued continuous functions on [0,1] ?

Correct option is D

The standard inner product on two real valued continuous functions is given by:

​$$\langle f, g \rangle = \int_{0}^{1} f(t)g(t) \, dt$$ . which is Option D.

And also , if  $$\langle f, g \rangle = \int_{0}^{1} f(t)g(t) \, dt$$ Then.

(i) Symmetric property :

​$$\langle f, g \rangle = \int_{0}^{1} f(t)g(t) \, dt\\=\int_{0}^{1} g(t)f(t)=\langle g,f \rangle\ \textbf{(satisfied)}$$

(ii) Linearty property : Let , $$a,b\ \in \R\ (Field)$$ then ,

​$$\langle af+bg, h \rangle = \int_{0}^{1} (af+bg)(t).h(t) \, dt\\= \int_{0}^{1} \big(af(t)+bg(t)\big).h(t)\\=\int_{0}^{1}\big[af(t).h(t)+bg(t).h(t)\big]dt\\=a\int_{0}^{1}f(t).h(t)dt+b\int_{0}^{1}g(t).h(t)dt\\=a\langle f,h\rangle+b\langle g,h\rangle\\\textbf{(Satisfied)}\\\text{As, both required properties are satisfied , it is an inner product}$$.So, $$\textbf{Option D is correct.}$$​​

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