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Tomasz Kania
Tomasz Kania
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The conjecture which you stated is false. A counterexample is contained in the proof of Figiel [Studia Math. 42 (1972), 295–306] [Studia Math. 42 (1972), 295–306]. He actually proves that squares of finite-dimensional subspaces of the space he constructs are not uniformly embeddable into the space itself.

I am unaware of a simpler counterexample for infinite-dimensional spaces.

The conjecture which you stated is false. A counterexample is contained in the proof of Figiel [Studia Math. 42 (1972), 295–306]. He actually proves that squares of finite-dimensional subspaces of the space he constructs are not uniformly embeddable into the space itself.

I am unaware of a simpler counterexample for infinite-dimensional spaces.

The conjecture which you stated is false. A counterexample is contained in the proof of Figiel [Studia Math. 42 (1972), 295–306]. He actually proves that squares of finite-dimensional subspaces of the space he constructs are not uniformly embeddable into the space itself.

I am unaware of a simpler counterexample for infinite-dimensional spaces.

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Mikhail Ostrovskii
Mikhail Ostrovskii
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The conjecture which you stated is false. A counterexample is contained in the proof of Figiel [Studia Math. 42 (1972), 295–306]. He actually proves that squares of finite-dimensional subspaces of the space he constructs are not uniformly embeddable into the space itself.

I am unaware of a simpler counterexample for infinite-dimensional spaces.