Answer to Question #150758 in Functional Analysis for Ashweta Padhan

Question #150758
Show that (C[a, b], ||f||2) is a norm
1
Expert's answer
2020-12-15T13:37:43-0500

First of all, for f∈C([a;b])f\in\mathcal{C}([a;b]) we see that

∣∣f∣∣2=(∫ab∣f(x)∣2dx)1/2||f||_2 = (\int_a^b |f(x)|^2 dx)^{1/2}

is well defined and is in [0;+∞)[0;+\infty) .

Now let's verify the norm properties:

  1. Separation, ∣∣f∣∣2=0↔f=0||f||_2 = 0 \leftrightarrow f=0 . If f=0,∣∣f∣∣2=0f=0, ||f||_2=0 is obvious. Now suppose fβ‰ 0,∣f∣2β‰ 0f\neq 0, |f|^2 \neq 0 is a continuous positive function and thus ∫ab∣f(x)∣2dxβ‰ 0\int_a^b |f(x)|^2 dx \neq 0, therefore ∣∣f∣∣2β‰ 0||f||_2\neq 0 .
  2. Linearity, ∣∣αf∣∣2=βˆ£Ξ±βˆ£β‹…βˆ£βˆ£f∣∣2||\alpha f||_2 = |\alpha| \cdot||f||_2 . It follows from linearity of an integral : (∫ab∣αf∣2(x)dx)1/2=(∣α∣2∫ab∣f(x)∣2dx)1/2=βˆ£Ξ±βˆ£β‹…βˆ£βˆ£f∣∣2(\int_a^b |\alpha f|^2(x) dx)^{1/2} = (|\alpha|^2 \int_a^b |f(x)|^2 dx)^{1/2}=|\alpha|\cdot||f||_2
  3. Triangle inequality, ∣∣f+g∣∣2β‰€βˆ£βˆ£f∣∣2+∣∣g∣∣2||f+g||_2 \leq ||f||_2 + ||g||_2 .

∫ab∣f+g∣2(x)dxβ‰€βˆ«ab∣f∣2(x)dx+∫ab∣g∣2(x)dx+2∫ab∣fg∣(x)dx\int_a^b|f+g|^2(x) dx \leq \int_a^b|f|^2(x) dx +\int_a^b|g|^2(x)dx + 2 \int_a^b |fg| (x) dx

We have (∫ab∣fg∣(x)dx)2≀(∫ab∣f(x)∣2dx)β‹…(∫ab∣g(x)∣2dx)(\int_a^b |fg|(x) dx)^2 \leq (\int_a^b |f(x)|^2 dx) \cdot (\int_a^b |g(x)|^2 dx), as

∣fg∣(∫ab∣f(x)∣2dx)1/2β‹…(∫ab∣g(x)∣2dx)1/2β‰€βˆ£f∣2∫ab∣f(x)∣2dx+∣g∣2∫ab∣g(x)∣2dx2\frac{|fg|}{(\int_a^b |f(x)|^2 dx)^{1/2}\cdot(\int_a^b |g(x)|^2 dx)^{1/2}} \leq \frac{\frac{|f|^2}{\int_a^b |f(x)|^2 dx}+\frac{|g|^2}{\int_a^b |g(x)|^2 dx}}{2} (by applying 2∣abβˆ£β‰€a2+b22|ab|\leq a^2+b^2 to ∣f∣(∫ab∣f(x)∣2dx)1/2\frac{|f|}{(\int_a^b |f(x)|^2 dx)^{1/2}} and ∣g∣(∫ab∣g(x)∣2dx)1/2\frac{|g|}{(\int_a^b |g(x)|^2 dx)^{1/2}} ) and thus by integrating on [a;b][a;b] we find :

∫ab∣fg∣(x)dx(∫ab∣f(x)∣2dx)1/2β‹…(∫ab∣g(x)∣2dx)1/2≀1\frac{\int_a^b|fg| (x)dx}{(\int_a^b |f(x)|^2dx)^{1/2}\cdot(\int_a^b |g(x)|^2 dx)^{1/2}} \leq 1

Therefore (∫ab∣fg∣(x)dx)2≀(∫ab∣f∣2dx)β‹…(∫ab∣g∣2dx)(\int_a^b |fg|(x) dx)^2 \leq (\int_a^b |f|^2 dx) \cdot (\int_a^b |g|^2 dx) .

And we conclude :

∣∣f+g∣∣22=∫ab∣f+g∣2β‰€βˆ«ab∣f∣2+∫ab∣g∣2+2∫ab∣fg∣||f+g||_2^2 = \int_a^b|f+g|^2 \leq \int_a^b|f|^2 +\int_a^b |g|^2 +2\int_a^b |fg|

∣∣f+g∣∣22β‰€βˆ«ab∣f∣2+∫ab∣g∣2+2(∫ab∣f∣2)1/2(∫ab∣g∣2)1/2||f+g||_2^2 \leq \int_a^b|f|^2 +\int_a^b|g|^2 +2(\int_a^b|f|^2)^{1/2}(\int_a^b|g|^2)^{1/2}

∣∣f+g∣∣22≀(∣∣f∣∣2+∣∣g∣∣2)2||f+g||_2^2 \leq (||f||_2+||g||_2)^2

And thus ∣∣f+g∣∣2β‰€βˆ£βˆ£f∣∣2+∣∣g∣∣2||f+g||_2 \leq ||f||_2 + ||g||_2 .




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