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Let $(H, (\cdot,\cdot)_1)$ be a Hilbert space. Suppose also that $(\cdot,\cdot)_2$ is an inner product on $H$ which is norm-equivalent with $(\cdot,\cdot)$.

Is it possible to write the second inner product in terms of the first?

For example, perhaps it is true that $$(a,b)_2 = (Ta,b)_1$$ for some operator $T$? If so what is known about $T$?

matt.w
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    Define an operator $T$ on $H$ exactly like you did: $(Ta, b)_1 = (a,b)_2$. By norm equivalence of the inner products and Riesz representation, $T$ is a well-defined bounded operator. In fact, $T$ is an invertible positive operator. One could say the space $(H, (\cdot,\cdot)_1)$ is "renormed" by $T$. – Michael May 30 '13 at 17:47
  • @Michael You should post that as an answer... – ˈjuː.zɚ79365 May 31 '13 at 11:26

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Define an operator $T$ on $H$ exactly like you did: $(Ta,b)_1=(a,b)_2$. By norm equivalence of the inner products and Riesz representation, $T$ is a well-defined bounded operator. In fact, $T$ is an invertible positive operator. One could say the space $(H,(⋅,⋅)_1)$ is "renormed" by $T$.

Michael
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