- #1

spaghetti3451

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## Homework Statement

(a) Show that the set of all square-integrable functions is a vector space. Is the set of all normalised functions a vector space?

(b) Show that the integral ##\int^{a}_{b} f(x)^{*} g(x) dx## satisfies the conditions for an inner product.

## Homework Equations

The main problem is to show that the sum of two square-integrable functions is itself square-integrable.

Use Schwarz inequality: ##| \int^{a}_{b} f(x)^{*} g(x) dx| \leq \sqrt{ \int^{a}_{b} |f(x)|^{2} dx \int^{a}_{b} |g(x)|^{2} dx}##

## The Attempt at a Solution

(a) Let ##f(x)## and ##g(x)## be square-integrable functions. Then,

##\int^{a}_{b} | \alpha f(x) + \beta g(x)|^2 dx##

##= \int^{a}_{b} [\ | \alpha f(x) |^2 + | \beta f(x) |^2 + \alpha^{*} f(x)^{*} \beta g(x) + \alpha f(x) \beta^{*} g(x)^{*} ] dx ##

##= |\alpha|^2 \int^{a}_{b} | f(x) |^2 dx + |\beta|^2 \int^{a}_{b} | f(x) |^2 dx + \alpha^{*} \beta \int^{a}_{b} f(x)^{*} g(x) dx + \alpha \beta^{*} \int^{a}_{b} f(x) g(x)^{*} dx ##

The first two terms are already square-integrable. So, all I have to do now is to show that the last two terms are square-integrable.

Can you suggest how I can use the Schwarz inequality over the last two terms?

P.S. : The last two terms are complex conjugates of each other, and the sum of two complex conjugates is a real number. Does it somehow prove that the last two terms form a square-integrable combination?

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