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Prove that the set of all sequences of real numbers is not a inner product space

The author gave me an outline of the proof. Let $L$ be the set of all sequences of real numbers. For each $i$, let $e_i$ be the sequence $x_1, x_2, \dots, x_i, \dots$ with $x_i = 1$ and $x_r = 0$ for $r\neq i$.

  1. Suppose that an inner product can be defined in $L$, so as to give us an inner product space. Show that for each $i$, and each $M_i>0$, there is an $\alpha_i$ such that $\|\alpha_ie_i\| > M_i$. (Here, as usual, $\|A\| = \sqrt{A\cdot A}$).

  2. Show that there is a sequence $\alpha_i, \alpha_2, \dots$ such that $\|\sum_{i = 1}^v\alpha_ie_i\| > i, \forall i$.

  3. Let $s = \alpha_1, \alpha_2, \ldots \in L$. Let $k = \|s\|$, let $s_v = \sum_{i = 1}^v\alpha_ie_i$, and let $t_v = s = s_v$, so that $s = s_v + t_v$. Show that the $\alpha_i$'s can be chosen so that $s_v - t_v \ge 0$. Now show that this whole situation is impossible.

Ben Grossmann
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Hector Gaxiola
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  • You have an outline of the proof, so where exactly are you stuck? – Ben Grossmann Jan 17 '20 at 07:51
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    Assuming Axiom of Choice every vector space over $\mathbb R$ or $\mathbb C$ has an inner product. See the answer by Christian Blatter in https://math.stackexchange.com/questions/247425/is-there-a-vector-space-that-cannot-be-an-inner-product-space/3174794#3174794 – Kavi Rama Murthy Jan 17 '20 at 07:55
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    @Hector Also, given that the statement that the author is telling you to prove is apparently false, it would be good to know where exactly you came across this. – Ben Grossmann Jan 17 '20 at 08:00
  • In the inequality in $2$, $i$ appears as a summation index on the left but free on the right (and the entire inequality is under the scope of a universal quantifier for $i$) -- that doesn't match; is there a typo? Perhaps it should read $\cdots\gt v, \forall v$? – joriki Jan 17 '20 at 08:02
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    @Hector are we missing information? In this context, does the set of all sequences have some sort of topology relating to pointwise convergence? – Ben Grossmann Jan 17 '20 at 08:03
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    Perhaps what is meant is that there is no inner product of the form $(v,w) = \sum a_i v_i w_i$ – Calvin Khor Jan 17 '20 at 08:03
  • A difficulty we face when a problem is taken from a book, but we are not told what book. This is clearly not "the set" of all sequences of real numbers. It is that set together with some additional structure. For example, addition and salar multiplication; but perhaps more, say a topology. – GEdgar Jan 17 '20 at 13:40
  • The book that I am using is Calculus by Edwin E Moise the complete edition. Sorry I did forget to write the definition for addition and scaler multiplication. Addition is defined as (x₁, x₂,...) + (y₁, y₂, … ) = (x₁+y₁, x₂+y₂, …) and scaler multiplication is defined as α(x₁, x₂, …) = (αx₁, αx₂, …). The part that I am struggling with is that we assume that there exists a definition that will give us an inner product space but from what I understand is that we should only use properties of the inner product space and not an explicit formula. Therefore its hard for me to show part 1,2, and 3. – Hector Gaxiola Jan 17 '20 at 18:25

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